MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  evlfcl Structured version   Visualization version   GIF version

Theorem evlfcl 18175
Description: The evaluation functor is a bifunctor (a two-argument functor) with the first parameter taking values in the set of functors 𝐢⟢𝐷, and the second parameter in 𝐷. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
evlfcl.e 𝐸 = (𝐢 evalF 𝐷)
evlfcl.q 𝑄 = (𝐢 FuncCat 𝐷)
evlfcl.c (πœ‘ β†’ 𝐢 ∈ Cat)
evlfcl.d (πœ‘ β†’ 𝐷 ∈ Cat)
Assertion
Ref Expression
evlfcl (πœ‘ β†’ 𝐸 ∈ ((𝑄 Γ—c 𝐢) Func 𝐷))

Proof of Theorem evlfcl
Dummy variables 𝑓 π‘Ž 𝑔 β„Ž π‘š 𝑛 𝑒 𝑣 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evlfcl.e . . . . 5 𝐸 = (𝐢 evalF 𝐷)
2 evlfcl.c . . . . 5 (πœ‘ β†’ 𝐢 ∈ Cat)
3 evlfcl.d . . . . 5 (πœ‘ β†’ 𝐷 ∈ Cat)
4 eqid 2733 . . . . 5 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
5 eqid 2733 . . . . 5 (Hom β€˜πΆ) = (Hom β€˜πΆ)
6 eqid 2733 . . . . 5 (compβ€˜π·) = (compβ€˜π·)
7 eqid 2733 . . . . 5 (𝐢 Nat 𝐷) = (𝐢 Nat 𝐷)
81, 2, 3, 4, 5, 6, 7evlfval 18170 . . . 4 (πœ‘ β†’ 𝐸 = ⟨(𝑓 ∈ (𝐢 Func 𝐷), π‘₯ ∈ (Baseβ€˜πΆ) ↦ ((1st β€˜π‘“)β€˜π‘₯)), (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)), 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ↦ ⦋(1st β€˜π‘₯) / π‘šβ¦Œβ¦‹(1st β€˜π‘¦) / π‘›β¦Œ(π‘Ž ∈ (π‘š(𝐢 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦)) ↦ ((π‘Žβ€˜(2nd β€˜π‘¦))(⟨((1st β€˜π‘š)β€˜(2nd β€˜π‘₯)), ((1st β€˜π‘š)β€˜(2nd β€˜π‘¦))⟩(compβ€˜π·)((1st β€˜π‘›)β€˜(2nd β€˜π‘¦)))(((2nd β€˜π‘₯)(2nd β€˜π‘š)(2nd β€˜π‘¦))β€˜π‘”))))⟩)
9 ovex 7442 . . . . . 6 (𝐢 Func 𝐷) ∈ V
10 fvex 6905 . . . . . 6 (Baseβ€˜πΆ) ∈ V
119, 10mpoex 8066 . . . . 5 (𝑓 ∈ (𝐢 Func 𝐷), π‘₯ ∈ (Baseβ€˜πΆ) ↦ ((1st β€˜π‘“)β€˜π‘₯)) ∈ V
129, 10xpex 7740 . . . . . 6 ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∈ V
1312, 12mpoex 8066 . . . . 5 (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)), 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ↦ ⦋(1st β€˜π‘₯) / π‘šβ¦Œβ¦‹(1st β€˜π‘¦) / π‘›β¦Œ(π‘Ž ∈ (π‘š(𝐢 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦)) ↦ ((π‘Žβ€˜(2nd β€˜π‘¦))(⟨((1st β€˜π‘š)β€˜(2nd β€˜π‘₯)), ((1st β€˜π‘š)β€˜(2nd β€˜π‘¦))⟩(compβ€˜π·)((1st β€˜π‘›)β€˜(2nd β€˜π‘¦)))(((2nd β€˜π‘₯)(2nd β€˜π‘š)(2nd β€˜π‘¦))β€˜π‘”)))) ∈ V
1411, 13opelvv 5717 . . . 4 ⟨(𝑓 ∈ (𝐢 Func 𝐷), π‘₯ ∈ (Baseβ€˜πΆ) ↦ ((1st β€˜π‘“)β€˜π‘₯)), (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)), 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ↦ ⦋(1st β€˜π‘₯) / π‘šβ¦Œβ¦‹(1st β€˜π‘¦) / π‘›β¦Œ(π‘Ž ∈ (π‘š(𝐢 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦)) ↦ ((π‘Žβ€˜(2nd β€˜π‘¦))(⟨((1st β€˜π‘š)β€˜(2nd β€˜π‘₯)), ((1st β€˜π‘š)β€˜(2nd β€˜π‘¦))⟩(compβ€˜π·)((1st β€˜π‘›)β€˜(2nd β€˜π‘¦)))(((2nd β€˜π‘₯)(2nd β€˜π‘š)(2nd β€˜π‘¦))β€˜π‘”))))⟩ ∈ (V Γ— V)
158, 14eqeltrdi 2842 . . 3 (πœ‘ β†’ 𝐸 ∈ (V Γ— V))
16 1st2nd2 8014 . . 3 (𝐸 ∈ (V Γ— V) β†’ 𝐸 = ⟨(1st β€˜πΈ), (2nd β€˜πΈ)⟩)
1715, 16syl 17 . 2 (πœ‘ β†’ 𝐸 = ⟨(1st β€˜πΈ), (2nd β€˜πΈ)⟩)
18 eqid 2733 . . . . 5 (𝑄 Γ—c 𝐢) = (𝑄 Γ—c 𝐢)
19 evlfcl.q . . . . . 6 𝑄 = (𝐢 FuncCat 𝐷)
2019fucbas 17912 . . . . 5 (𝐢 Func 𝐷) = (Baseβ€˜π‘„)
2118, 20, 4xpcbas 18130 . . . 4 ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) = (Baseβ€˜(𝑄 Γ—c 𝐢))
22 eqid 2733 . . . 4 (Baseβ€˜π·) = (Baseβ€˜π·)
23 eqid 2733 . . . 4 (Hom β€˜(𝑄 Γ—c 𝐢)) = (Hom β€˜(𝑄 Γ—c 𝐢))
24 eqid 2733 . . . 4 (Hom β€˜π·) = (Hom β€˜π·)
25 eqid 2733 . . . 4 (Idβ€˜(𝑄 Γ—c 𝐢)) = (Idβ€˜(𝑄 Γ—c 𝐢))
26 eqid 2733 . . . 4 (Idβ€˜π·) = (Idβ€˜π·)
27 eqid 2733 . . . 4 (compβ€˜(𝑄 Γ—c 𝐢)) = (compβ€˜(𝑄 Γ—c 𝐢))
2819, 2, 3fuccat 17923 . . . . 5 (πœ‘ β†’ 𝑄 ∈ Cat)
2918, 28, 2xpccat 18142 . . . 4 (πœ‘ β†’ (𝑄 Γ—c 𝐢) ∈ Cat)
30 relfunc 17812 . . . . . . . . . . 11 Rel (𝐢 Func 𝐷)
31 simpr 486 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑓 ∈ (𝐢 Func 𝐷)) β†’ 𝑓 ∈ (𝐢 Func 𝐷))
32 1st2ndbr 8028 . . . . . . . . . . 11 ((Rel (𝐢 Func 𝐷) ∧ 𝑓 ∈ (𝐢 Func 𝐷)) β†’ (1st β€˜π‘“)(𝐢 Func 𝐷)(2nd β€˜π‘“))
3330, 31, 32sylancr 588 . . . . . . . . . 10 ((πœ‘ ∧ 𝑓 ∈ (𝐢 Func 𝐷)) β†’ (1st β€˜π‘“)(𝐢 Func 𝐷)(2nd β€˜π‘“))
344, 22, 33funcf1 17816 . . . . . . . . 9 ((πœ‘ ∧ 𝑓 ∈ (𝐢 Func 𝐷)) β†’ (1st β€˜π‘“):(Baseβ€˜πΆ)⟢(Baseβ€˜π·))
3534ffvelcdmda 7087 . . . . . . . 8 (((πœ‘ ∧ 𝑓 ∈ (𝐢 Func 𝐷)) ∧ π‘₯ ∈ (Baseβ€˜πΆ)) β†’ ((1st β€˜π‘“)β€˜π‘₯) ∈ (Baseβ€˜π·))
3635ralrimiva 3147 . . . . . . 7 ((πœ‘ ∧ 𝑓 ∈ (𝐢 Func 𝐷)) β†’ βˆ€π‘₯ ∈ (Baseβ€˜πΆ)((1st β€˜π‘“)β€˜π‘₯) ∈ (Baseβ€˜π·))
3736ralrimiva 3147 . . . . . 6 (πœ‘ β†’ βˆ€π‘“ ∈ (𝐢 Func 𝐷)βˆ€π‘₯ ∈ (Baseβ€˜πΆ)((1st β€˜π‘“)β€˜π‘₯) ∈ (Baseβ€˜π·))
38 eqid 2733 . . . . . . 7 (𝑓 ∈ (𝐢 Func 𝐷), π‘₯ ∈ (Baseβ€˜πΆ) ↦ ((1st β€˜π‘“)β€˜π‘₯)) = (𝑓 ∈ (𝐢 Func 𝐷), π‘₯ ∈ (Baseβ€˜πΆ) ↦ ((1st β€˜π‘“)β€˜π‘₯))
3938fmpo 8054 . . . . . 6 (βˆ€π‘“ ∈ (𝐢 Func 𝐷)βˆ€π‘₯ ∈ (Baseβ€˜πΆ)((1st β€˜π‘“)β€˜π‘₯) ∈ (Baseβ€˜π·) ↔ (𝑓 ∈ (𝐢 Func 𝐷), π‘₯ ∈ (Baseβ€˜πΆ) ↦ ((1st β€˜π‘“)β€˜π‘₯)):((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))⟢(Baseβ€˜π·))
4037, 39sylib 217 . . . . 5 (πœ‘ β†’ (𝑓 ∈ (𝐢 Func 𝐷), π‘₯ ∈ (Baseβ€˜πΆ) ↦ ((1st β€˜π‘“)β€˜π‘₯)):((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))⟢(Baseβ€˜π·))
4111, 13op1std 7985 . . . . . . 7 (𝐸 = ⟨(𝑓 ∈ (𝐢 Func 𝐷), π‘₯ ∈ (Baseβ€˜πΆ) ↦ ((1st β€˜π‘“)β€˜π‘₯)), (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)), 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ↦ ⦋(1st β€˜π‘₯) / π‘šβ¦Œβ¦‹(1st β€˜π‘¦) / π‘›β¦Œ(π‘Ž ∈ (π‘š(𝐢 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦)) ↦ ((π‘Žβ€˜(2nd β€˜π‘¦))(⟨((1st β€˜π‘š)β€˜(2nd β€˜π‘₯)), ((1st β€˜π‘š)β€˜(2nd β€˜π‘¦))⟩(compβ€˜π·)((1st β€˜π‘›)β€˜(2nd β€˜π‘¦)))(((2nd β€˜π‘₯)(2nd β€˜π‘š)(2nd β€˜π‘¦))β€˜π‘”))))⟩ β†’ (1st β€˜πΈ) = (𝑓 ∈ (𝐢 Func 𝐷), π‘₯ ∈ (Baseβ€˜πΆ) ↦ ((1st β€˜π‘“)β€˜π‘₯)))
428, 41syl 17 . . . . . 6 (πœ‘ β†’ (1st β€˜πΈ) = (𝑓 ∈ (𝐢 Func 𝐷), π‘₯ ∈ (Baseβ€˜πΆ) ↦ ((1st β€˜π‘“)β€˜π‘₯)))
4342feq1d 6703 . . . . 5 (πœ‘ β†’ ((1st β€˜πΈ):((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))⟢(Baseβ€˜π·) ↔ (𝑓 ∈ (𝐢 Func 𝐷), π‘₯ ∈ (Baseβ€˜πΆ) ↦ ((1st β€˜π‘“)β€˜π‘₯)):((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))⟢(Baseβ€˜π·)))
4440, 43mpbird 257 . . . 4 (πœ‘ β†’ (1st β€˜πΈ):((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))⟢(Baseβ€˜π·))
45 eqid 2733 . . . . . 6 (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)), 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ↦ ⦋(1st β€˜π‘₯) / π‘šβ¦Œβ¦‹(1st β€˜π‘¦) / π‘›β¦Œ(π‘Ž ∈ (π‘š(𝐢 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦)) ↦ ((π‘Žβ€˜(2nd β€˜π‘¦))(⟨((1st β€˜π‘š)β€˜(2nd β€˜π‘₯)), ((1st β€˜π‘š)β€˜(2nd β€˜π‘¦))⟩(compβ€˜π·)((1st β€˜π‘›)β€˜(2nd β€˜π‘¦)))(((2nd β€˜π‘₯)(2nd β€˜π‘š)(2nd β€˜π‘¦))β€˜π‘”)))) = (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)), 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ↦ ⦋(1st β€˜π‘₯) / π‘šβ¦Œβ¦‹(1st β€˜π‘¦) / π‘›β¦Œ(π‘Ž ∈ (π‘š(𝐢 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦)) ↦ ((π‘Žβ€˜(2nd β€˜π‘¦))(⟨((1st β€˜π‘š)β€˜(2nd β€˜π‘₯)), ((1st β€˜π‘š)β€˜(2nd β€˜π‘¦))⟩(compβ€˜π·)((1st β€˜π‘›)β€˜(2nd β€˜π‘¦)))(((2nd β€˜π‘₯)(2nd β€˜π‘š)(2nd β€˜π‘¦))β€˜π‘”))))
46 ovex 7442 . . . . . . . . 9 (π‘š(𝐢 Nat 𝐷)𝑛) ∈ V
47 ovex 7442 . . . . . . . . 9 ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦)) ∈ V
4846, 47mpoex 8066 . . . . . . . 8 (π‘Ž ∈ (π‘š(𝐢 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦)) ↦ ((π‘Žβ€˜(2nd β€˜π‘¦))(⟨((1st β€˜π‘š)β€˜(2nd β€˜π‘₯)), ((1st β€˜π‘š)β€˜(2nd β€˜π‘¦))⟩(compβ€˜π·)((1st β€˜π‘›)β€˜(2nd β€˜π‘¦)))(((2nd β€˜π‘₯)(2nd β€˜π‘š)(2nd β€˜π‘¦))β€˜π‘”))) ∈ V
4948csbex 5312 . . . . . . 7 ⦋(1st β€˜π‘¦) / π‘›β¦Œ(π‘Ž ∈ (π‘š(𝐢 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦)) ↦ ((π‘Žβ€˜(2nd β€˜π‘¦))(⟨((1st β€˜π‘š)β€˜(2nd β€˜π‘₯)), ((1st β€˜π‘š)β€˜(2nd β€˜π‘¦))⟩(compβ€˜π·)((1st β€˜π‘›)β€˜(2nd β€˜π‘¦)))(((2nd β€˜π‘₯)(2nd β€˜π‘š)(2nd β€˜π‘¦))β€˜π‘”))) ∈ V
5049csbex 5312 . . . . . 6 ⦋(1st β€˜π‘₯) / π‘šβ¦Œβ¦‹(1st β€˜π‘¦) / π‘›β¦Œ(π‘Ž ∈ (π‘š(𝐢 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦)) ↦ ((π‘Žβ€˜(2nd β€˜π‘¦))(⟨((1st β€˜π‘š)β€˜(2nd β€˜π‘₯)), ((1st β€˜π‘š)β€˜(2nd β€˜π‘¦))⟩(compβ€˜π·)((1st β€˜π‘›)β€˜(2nd β€˜π‘¦)))(((2nd β€˜π‘₯)(2nd β€˜π‘š)(2nd β€˜π‘¦))β€˜π‘”))) ∈ V
5145, 50fnmpoi 8056 . . . . 5 (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)), 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ↦ ⦋(1st β€˜π‘₯) / π‘šβ¦Œβ¦‹(1st β€˜π‘¦) / π‘›β¦Œ(π‘Ž ∈ (π‘š(𝐢 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦)) ↦ ((π‘Žβ€˜(2nd β€˜π‘¦))(⟨((1st β€˜π‘š)β€˜(2nd β€˜π‘₯)), ((1st β€˜π‘š)β€˜(2nd β€˜π‘¦))⟩(compβ€˜π·)((1st β€˜π‘›)β€˜(2nd β€˜π‘¦)))(((2nd β€˜π‘₯)(2nd β€˜π‘š)(2nd β€˜π‘¦))β€˜π‘”)))) Fn (((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) Γ— ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)))
5211, 13op2ndd 7986 . . . . . . 7 (𝐸 = ⟨(𝑓 ∈ (𝐢 Func 𝐷), π‘₯ ∈ (Baseβ€˜πΆ) ↦ ((1st β€˜π‘“)β€˜π‘₯)), (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)), 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ↦ ⦋(1st β€˜π‘₯) / π‘šβ¦Œβ¦‹(1st β€˜π‘¦) / π‘›β¦Œ(π‘Ž ∈ (π‘š(𝐢 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦)) ↦ ((π‘Žβ€˜(2nd β€˜π‘¦))(⟨((1st β€˜π‘š)β€˜(2nd β€˜π‘₯)), ((1st β€˜π‘š)β€˜(2nd β€˜π‘¦))⟩(compβ€˜π·)((1st β€˜π‘›)β€˜(2nd β€˜π‘¦)))(((2nd β€˜π‘₯)(2nd β€˜π‘š)(2nd β€˜π‘¦))β€˜π‘”))))⟩ β†’ (2nd β€˜πΈ) = (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)), 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ↦ ⦋(1st β€˜π‘₯) / π‘šβ¦Œβ¦‹(1st β€˜π‘¦) / π‘›β¦Œ(π‘Ž ∈ (π‘š(𝐢 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦)) ↦ ((π‘Žβ€˜(2nd β€˜π‘¦))(⟨((1st β€˜π‘š)β€˜(2nd β€˜π‘₯)), ((1st β€˜π‘š)β€˜(2nd β€˜π‘¦))⟩(compβ€˜π·)((1st β€˜π‘›)β€˜(2nd β€˜π‘¦)))(((2nd β€˜π‘₯)(2nd β€˜π‘š)(2nd β€˜π‘¦))β€˜π‘”)))))
538, 52syl 17 . . . . . 6 (πœ‘ β†’ (2nd β€˜πΈ) = (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)), 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ↦ ⦋(1st β€˜π‘₯) / π‘šβ¦Œβ¦‹(1st β€˜π‘¦) / π‘›β¦Œ(π‘Ž ∈ (π‘š(𝐢 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦)) ↦ ((π‘Žβ€˜(2nd β€˜π‘¦))(⟨((1st β€˜π‘š)β€˜(2nd β€˜π‘₯)), ((1st β€˜π‘š)β€˜(2nd β€˜π‘¦))⟩(compβ€˜π·)((1st β€˜π‘›)β€˜(2nd β€˜π‘¦)))(((2nd β€˜π‘₯)(2nd β€˜π‘š)(2nd β€˜π‘¦))β€˜π‘”)))))
5453fneq1d 6643 . . . . 5 (πœ‘ β†’ ((2nd β€˜πΈ) Fn (((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) Γ— ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ↔ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)), 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ↦ ⦋(1st β€˜π‘₯) / π‘šβ¦Œβ¦‹(1st β€˜π‘¦) / π‘›β¦Œ(π‘Ž ∈ (π‘š(𝐢 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦)) ↦ ((π‘Žβ€˜(2nd β€˜π‘¦))(⟨((1st β€˜π‘š)β€˜(2nd β€˜π‘₯)), ((1st β€˜π‘š)β€˜(2nd β€˜π‘¦))⟩(compβ€˜π·)((1st β€˜π‘›)β€˜(2nd β€˜π‘¦)))(((2nd β€˜π‘₯)(2nd β€˜π‘š)(2nd β€˜π‘¦))β€˜π‘”)))) Fn (((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) Γ— ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)))))
5551, 54mpbiri 258 . . . 4 (πœ‘ β†’ (2nd β€˜πΈ) Fn (((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) Γ— ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))))
563ad2antrr 725 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) β†’ 𝐷 ∈ Cat)
5756adantr 482 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) ∧ (π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔) ∧ β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣))) β†’ 𝐷 ∈ Cat)
58 simplrl 776 . . . . . . . . . . . . . . . . . . 19 (((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) β†’ 𝑓 ∈ (𝐢 Func 𝐷))
5930, 58, 32sylancr 588 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) β†’ (1st β€˜π‘“)(𝐢 Func 𝐷)(2nd β€˜π‘“))
604, 22, 59funcf1 17816 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) β†’ (1st β€˜π‘“):(Baseβ€˜πΆ)⟢(Baseβ€˜π·))
6160adantr 482 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) ∧ (π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔) ∧ β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣))) β†’ (1st β€˜π‘“):(Baseβ€˜πΆ)⟢(Baseβ€˜π·))
62 simplrr 777 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) β†’ 𝑒 ∈ (Baseβ€˜πΆ))
6362adantr 482 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) ∧ (π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔) ∧ β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣))) β†’ 𝑒 ∈ (Baseβ€˜πΆ))
6461, 63ffvelcdmd 7088 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) ∧ (π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔) ∧ β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣))) β†’ ((1st β€˜π‘“)β€˜π‘’) ∈ (Baseβ€˜π·))
65 simplrr 777 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) ∧ (π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔) ∧ β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣))) β†’ 𝑣 ∈ (Baseβ€˜πΆ))
6661, 65ffvelcdmd 7088 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) ∧ (π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔) ∧ β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣))) β†’ ((1st β€˜π‘“)β€˜π‘£) ∈ (Baseβ€˜π·))
67 simprl 770 . . . . . . . . . . . . . . . . . . 19 (((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) β†’ 𝑔 ∈ (𝐢 Func 𝐷))
68 1st2ndbr 8028 . . . . . . . . . . . . . . . . . . 19 ((Rel (𝐢 Func 𝐷) ∧ 𝑔 ∈ (𝐢 Func 𝐷)) β†’ (1st β€˜π‘”)(𝐢 Func 𝐷)(2nd β€˜π‘”))
6930, 67, 68sylancr 588 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) β†’ (1st β€˜π‘”)(𝐢 Func 𝐷)(2nd β€˜π‘”))
704, 22, 69funcf1 17816 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) β†’ (1st β€˜π‘”):(Baseβ€˜πΆ)⟢(Baseβ€˜π·))
7170adantr 482 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) ∧ (π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔) ∧ β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣))) β†’ (1st β€˜π‘”):(Baseβ€˜πΆ)⟢(Baseβ€˜π·))
7271, 65ffvelcdmd 7088 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) ∧ (π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔) ∧ β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣))) β†’ ((1st β€˜π‘”)β€˜π‘£) ∈ (Baseβ€˜π·))
73 simprr 772 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) β†’ 𝑣 ∈ (Baseβ€˜πΆ))
744, 5, 24, 59, 62, 73funcf2 17818 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) β†’ (𝑒(2nd β€˜π‘“)𝑣):(𝑒(Hom β€˜πΆ)𝑣)⟢(((1st β€˜π‘“)β€˜π‘’)(Hom β€˜π·)((1st β€˜π‘“)β€˜π‘£)))
7574adantr 482 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) ∧ (π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔) ∧ β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣))) β†’ (𝑒(2nd β€˜π‘“)𝑣):(𝑒(Hom β€˜πΆ)𝑣)⟢(((1st β€˜π‘“)β€˜π‘’)(Hom β€˜π·)((1st β€˜π‘“)β€˜π‘£)))
76 simprr 772 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) ∧ (π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔) ∧ β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣))) β†’ β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣))
7775, 76ffvelcdmd 7088 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) ∧ (π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔) ∧ β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣))) β†’ ((𝑒(2nd β€˜π‘“)𝑣)β€˜β„Ž) ∈ (((1st β€˜π‘“)β€˜π‘’)(Hom β€˜π·)((1st β€˜π‘“)β€˜π‘£)))
78 simprl 770 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) ∧ (π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔) ∧ β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣))) β†’ π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔))
797, 78nat1st2nd 17902 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) ∧ (π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔) ∧ β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣))) β†’ π‘Ž ∈ (⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩(𝐢 Nat 𝐷)⟨(1st β€˜π‘”), (2nd β€˜π‘”)⟩))
807, 79, 4, 24, 65natcl 17904 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) ∧ (π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔) ∧ β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣))) β†’ (π‘Žβ€˜π‘£) ∈ (((1st β€˜π‘“)β€˜π‘£)(Hom β€˜π·)((1st β€˜π‘”)β€˜π‘£)))
8122, 24, 6, 57, 64, 66, 72, 77, 80catcocl 17629 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) ∧ (π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔) ∧ β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣))) β†’ ((π‘Žβ€˜π‘£)(⟨((1st β€˜π‘“)β€˜π‘’), ((1st β€˜π‘“)β€˜π‘£)⟩(compβ€˜π·)((1st β€˜π‘”)β€˜π‘£))((𝑒(2nd β€˜π‘“)𝑣)β€˜β„Ž)) ∈ (((1st β€˜π‘“)β€˜π‘’)(Hom β€˜π·)((1st β€˜π‘”)β€˜π‘£)))
8281ralrimivva 3201 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) β†’ βˆ€π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔)βˆ€β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣)((π‘Žβ€˜π‘£)(⟨((1st β€˜π‘“)β€˜π‘’), ((1st β€˜π‘“)β€˜π‘£)⟩(compβ€˜π·)((1st β€˜π‘”)β€˜π‘£))((𝑒(2nd β€˜π‘“)𝑣)β€˜β„Ž)) ∈ (((1st β€˜π‘“)β€˜π‘’)(Hom β€˜π·)((1st β€˜π‘”)β€˜π‘£)))
83 eqid 2733 . . . . . . . . . . . . . 14 (π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔), β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣) ↦ ((π‘Žβ€˜π‘£)(⟨((1st β€˜π‘“)β€˜π‘’), ((1st β€˜π‘“)β€˜π‘£)⟩(compβ€˜π·)((1st β€˜π‘”)β€˜π‘£))((𝑒(2nd β€˜π‘“)𝑣)β€˜β„Ž))) = (π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔), β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣) ↦ ((π‘Žβ€˜π‘£)(⟨((1st β€˜π‘“)β€˜π‘’), ((1st β€˜π‘“)β€˜π‘£)⟩(compβ€˜π·)((1st β€˜π‘”)β€˜π‘£))((𝑒(2nd β€˜π‘“)𝑣)β€˜β„Ž)))
8483fmpo 8054 . . . . . . . . . . . . 13 (βˆ€π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔)βˆ€β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣)((π‘Žβ€˜π‘£)(⟨((1st β€˜π‘“)β€˜π‘’), ((1st β€˜π‘“)β€˜π‘£)⟩(compβ€˜π·)((1st β€˜π‘”)β€˜π‘£))((𝑒(2nd β€˜π‘“)𝑣)β€˜β„Ž)) ∈ (((1st β€˜π‘“)β€˜π‘’)(Hom β€˜π·)((1st β€˜π‘”)β€˜π‘£)) ↔ (π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔), β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣) ↦ ((π‘Žβ€˜π‘£)(⟨((1st β€˜π‘“)β€˜π‘’), ((1st β€˜π‘“)β€˜π‘£)⟩(compβ€˜π·)((1st β€˜π‘”)β€˜π‘£))((𝑒(2nd β€˜π‘“)𝑣)β€˜β„Ž))):((𝑓(𝐢 Nat 𝐷)𝑔) Γ— (𝑒(Hom β€˜πΆ)𝑣))⟢(((1st β€˜π‘“)β€˜π‘’)(Hom β€˜π·)((1st β€˜π‘”)β€˜π‘£)))
8582, 84sylib 217 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) β†’ (π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔), β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣) ↦ ((π‘Žβ€˜π‘£)(⟨((1st β€˜π‘“)β€˜π‘’), ((1st β€˜π‘“)β€˜π‘£)⟩(compβ€˜π·)((1st β€˜π‘”)β€˜π‘£))((𝑒(2nd β€˜π‘“)𝑣)β€˜β„Ž))):((𝑓(𝐢 Nat 𝐷)𝑔) Γ— (𝑒(Hom β€˜πΆ)𝑣))⟢(((1st β€˜π‘“)β€˜π‘’)(Hom β€˜π·)((1st β€˜π‘”)β€˜π‘£)))
862ad2antrr 725 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) β†’ 𝐢 ∈ Cat)
87 eqid 2733 . . . . . . . . . . . . . 14 (βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘”, π‘£βŸ©) = (βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘”, π‘£βŸ©)
881, 86, 56, 4, 5, 6, 7, 58, 67, 62, 73, 87evlf2 18171 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) β†’ (βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘”, π‘£βŸ©) = (π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔), β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣) ↦ ((π‘Žβ€˜π‘£)(⟨((1st β€˜π‘“)β€˜π‘’), ((1st β€˜π‘“)β€˜π‘£)⟩(compβ€˜π·)((1st β€˜π‘”)β€˜π‘£))((𝑒(2nd β€˜π‘“)𝑣)β€˜β„Ž))))
8988feq1d 6703 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) β†’ ((βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘”, π‘£βŸ©):((𝑓(𝐢 Nat 𝐷)𝑔) Γ— (𝑒(Hom β€˜πΆ)𝑣))⟢(((1st β€˜π‘“)β€˜π‘’)(Hom β€˜π·)((1st β€˜π‘”)β€˜π‘£)) ↔ (π‘Ž ∈ (𝑓(𝐢 Nat 𝐷)𝑔), β„Ž ∈ (𝑒(Hom β€˜πΆ)𝑣) ↦ ((π‘Žβ€˜π‘£)(⟨((1st β€˜π‘“)β€˜π‘’), ((1st β€˜π‘“)β€˜π‘£)⟩(compβ€˜π·)((1st β€˜π‘”)β€˜π‘£))((𝑒(2nd β€˜π‘“)𝑣)β€˜β„Ž))):((𝑓(𝐢 Nat 𝐷)𝑔) Γ— (𝑒(Hom β€˜πΆ)𝑣))⟢(((1st β€˜π‘“)β€˜π‘’)(Hom β€˜π·)((1st β€˜π‘”)β€˜π‘£))))
9085, 89mpbird 257 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) β†’ (βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘”, π‘£βŸ©):((𝑓(𝐢 Nat 𝐷)𝑔) Γ— (𝑒(Hom β€˜πΆ)𝑣))⟢(((1st β€˜π‘“)β€˜π‘’)(Hom β€˜π·)((1st β€˜π‘”)β€˜π‘£)))
9119, 7fuchom 17913 . . . . . . . . . . . . 13 (𝐢 Nat 𝐷) = (Hom β€˜π‘„)
9218, 20, 4, 91, 5, 58, 62, 67, 73, 23xpchom2 18138 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) β†’ (βŸ¨π‘“, π‘’βŸ©(Hom β€˜(𝑄 Γ—c 𝐢))βŸ¨π‘”, π‘£βŸ©) = ((𝑓(𝐢 Nat 𝐷)𝑔) Γ— (𝑒(Hom β€˜πΆ)𝑣)))
931, 86, 56, 4, 58, 62evlf1 18173 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) β†’ (𝑓(1st β€˜πΈ)𝑒) = ((1st β€˜π‘“)β€˜π‘’))
941, 86, 56, 4, 67, 73evlf1 18173 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) β†’ (𝑔(1st β€˜πΈ)𝑣) = ((1st β€˜π‘”)β€˜π‘£))
9593, 94oveq12d 7427 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) β†’ ((𝑓(1st β€˜πΈ)𝑒)(Hom β€˜π·)(𝑔(1st β€˜πΈ)𝑣)) = (((1st β€˜π‘“)β€˜π‘’)(Hom β€˜π·)((1st β€˜π‘”)β€˜π‘£)))
9692, 95feq23d 6713 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) β†’ ((βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘”, π‘£βŸ©):(βŸ¨π‘“, π‘’βŸ©(Hom β€˜(𝑄 Γ—c 𝐢))βŸ¨π‘”, π‘£βŸ©)⟢((𝑓(1st β€˜πΈ)𝑒)(Hom β€˜π·)(𝑔(1st β€˜πΈ)𝑣)) ↔ (βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘”, π‘£βŸ©):((𝑓(𝐢 Nat 𝐷)𝑔) Γ— (𝑒(Hom β€˜πΆ)𝑣))⟢(((1st β€˜π‘“)β€˜π‘’)(Hom β€˜π·)((1st β€˜π‘”)β€˜π‘£))))
9790, 96mpbird 257 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) ∧ (𝑔 ∈ (𝐢 Func 𝐷) ∧ 𝑣 ∈ (Baseβ€˜πΆ))) β†’ (βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘”, π‘£βŸ©):(βŸ¨π‘“, π‘’βŸ©(Hom β€˜(𝑄 Γ—c 𝐢))βŸ¨π‘”, π‘£βŸ©)⟢((𝑓(1st β€˜πΈ)𝑒)(Hom β€˜π·)(𝑔(1st β€˜πΈ)𝑣)))
9897ralrimivva 3201 . . . . . . . . 9 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ βˆ€π‘” ∈ (𝐢 Func 𝐷)βˆ€π‘£ ∈ (Baseβ€˜πΆ)(βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘”, π‘£βŸ©):(βŸ¨π‘“, π‘’βŸ©(Hom β€˜(𝑄 Γ—c 𝐢))βŸ¨π‘”, π‘£βŸ©)⟢((𝑓(1st β€˜πΈ)𝑒)(Hom β€˜π·)(𝑔(1st β€˜πΈ)𝑣)))
9998ralrimivva 3201 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘“ ∈ (𝐢 Func 𝐷)βˆ€π‘’ ∈ (Baseβ€˜πΆ)βˆ€π‘” ∈ (𝐢 Func 𝐷)βˆ€π‘£ ∈ (Baseβ€˜πΆ)(βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘”, π‘£βŸ©):(βŸ¨π‘“, π‘’βŸ©(Hom β€˜(𝑄 Γ—c 𝐢))βŸ¨π‘”, π‘£βŸ©)⟢((𝑓(1st β€˜πΈ)𝑒)(Hom β€˜π·)(𝑔(1st β€˜πΈ)𝑣)))
100 oveq2 7417 . . . . . . . . . . . 12 (𝑦 = βŸ¨π‘”, π‘£βŸ© β†’ (π‘₯(2nd β€˜πΈ)𝑦) = (π‘₯(2nd β€˜πΈ)βŸ¨π‘”, π‘£βŸ©))
101 oveq2 7417 . . . . . . . . . . . 12 (𝑦 = βŸ¨π‘”, π‘£βŸ© β†’ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) = (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))βŸ¨π‘”, π‘£βŸ©))
102 fveq2 6892 . . . . . . . . . . . . . 14 (𝑦 = βŸ¨π‘”, π‘£βŸ© β†’ ((1st β€˜πΈ)β€˜π‘¦) = ((1st β€˜πΈ)β€˜βŸ¨π‘”, π‘£βŸ©))
103 df-ov 7412 . . . . . . . . . . . . . 14 (𝑔(1st β€˜πΈ)𝑣) = ((1st β€˜πΈ)β€˜βŸ¨π‘”, π‘£βŸ©)
104102, 103eqtr4di 2791 . . . . . . . . . . . . 13 (𝑦 = βŸ¨π‘”, π‘£βŸ© β†’ ((1st β€˜πΈ)β€˜π‘¦) = (𝑔(1st β€˜πΈ)𝑣))
105104oveq2d 7425 . . . . . . . . . . . 12 (𝑦 = βŸ¨π‘”, π‘£βŸ© β†’ (((1st β€˜πΈ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΈ)β€˜π‘¦)) = (((1st β€˜πΈ)β€˜π‘₯)(Hom β€˜π·)(𝑔(1st β€˜πΈ)𝑣)))
106100, 101, 105feq123d 6707 . . . . . . . . . . 11 (𝑦 = βŸ¨π‘”, π‘£βŸ© β†’ ((π‘₯(2nd β€˜πΈ)𝑦):(π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦)⟢(((1st β€˜πΈ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΈ)β€˜π‘¦)) ↔ (π‘₯(2nd β€˜πΈ)βŸ¨π‘”, π‘£βŸ©):(π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))βŸ¨π‘”, π‘£βŸ©)⟢(((1st β€˜πΈ)β€˜π‘₯)(Hom β€˜π·)(𝑔(1st β€˜πΈ)𝑣))))
107106ralxp 5842 . . . . . . . . . 10 (βˆ€π‘¦ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))(π‘₯(2nd β€˜πΈ)𝑦):(π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦)⟢(((1st β€˜πΈ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΈ)β€˜π‘¦)) ↔ βˆ€π‘” ∈ (𝐢 Func 𝐷)βˆ€π‘£ ∈ (Baseβ€˜πΆ)(π‘₯(2nd β€˜πΈ)βŸ¨π‘”, π‘£βŸ©):(π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))βŸ¨π‘”, π‘£βŸ©)⟢(((1st β€˜πΈ)β€˜π‘₯)(Hom β€˜π·)(𝑔(1st β€˜πΈ)𝑣)))
108 oveq1 7416 . . . . . . . . . . . 12 (π‘₯ = βŸ¨π‘“, π‘’βŸ© β†’ (π‘₯(2nd β€˜πΈ)βŸ¨π‘”, π‘£βŸ©) = (βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘”, π‘£βŸ©))
109 oveq1 7416 . . . . . . . . . . . 12 (π‘₯ = βŸ¨π‘“, π‘’βŸ© β†’ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))βŸ¨π‘”, π‘£βŸ©) = (βŸ¨π‘“, π‘’βŸ©(Hom β€˜(𝑄 Γ—c 𝐢))βŸ¨π‘”, π‘£βŸ©))
110 fveq2 6892 . . . . . . . . . . . . . 14 (π‘₯ = βŸ¨π‘“, π‘’βŸ© β†’ ((1st β€˜πΈ)β€˜π‘₯) = ((1st β€˜πΈ)β€˜βŸ¨π‘“, π‘’βŸ©))
111 df-ov 7412 . . . . . . . . . . . . . 14 (𝑓(1st β€˜πΈ)𝑒) = ((1st β€˜πΈ)β€˜βŸ¨π‘“, π‘’βŸ©)
112110, 111eqtr4di 2791 . . . . . . . . . . . . 13 (π‘₯ = βŸ¨π‘“, π‘’βŸ© β†’ ((1st β€˜πΈ)β€˜π‘₯) = (𝑓(1st β€˜πΈ)𝑒))
113112oveq1d 7424 . . . . . . . . . . . 12 (π‘₯ = βŸ¨π‘“, π‘’βŸ© β†’ (((1st β€˜πΈ)β€˜π‘₯)(Hom β€˜π·)(𝑔(1st β€˜πΈ)𝑣)) = ((𝑓(1st β€˜πΈ)𝑒)(Hom β€˜π·)(𝑔(1st β€˜πΈ)𝑣)))
114108, 109, 113feq123d 6707 . . . . . . . . . . 11 (π‘₯ = βŸ¨π‘“, π‘’βŸ© β†’ ((π‘₯(2nd β€˜πΈ)βŸ¨π‘”, π‘£βŸ©):(π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))βŸ¨π‘”, π‘£βŸ©)⟢(((1st β€˜πΈ)β€˜π‘₯)(Hom β€˜π·)(𝑔(1st β€˜πΈ)𝑣)) ↔ (βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘”, π‘£βŸ©):(βŸ¨π‘“, π‘’βŸ©(Hom β€˜(𝑄 Γ—c 𝐢))βŸ¨π‘”, π‘£βŸ©)⟢((𝑓(1st β€˜πΈ)𝑒)(Hom β€˜π·)(𝑔(1st β€˜πΈ)𝑣))))
1151142ralbidv 3219 . . . . . . . . . 10 (π‘₯ = βŸ¨π‘“, π‘’βŸ© β†’ (βˆ€π‘” ∈ (𝐢 Func 𝐷)βˆ€π‘£ ∈ (Baseβ€˜πΆ)(π‘₯(2nd β€˜πΈ)βŸ¨π‘”, π‘£βŸ©):(π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))βŸ¨π‘”, π‘£βŸ©)⟢(((1st β€˜πΈ)β€˜π‘₯)(Hom β€˜π·)(𝑔(1st β€˜πΈ)𝑣)) ↔ βˆ€π‘” ∈ (𝐢 Func 𝐷)βˆ€π‘£ ∈ (Baseβ€˜πΆ)(βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘”, π‘£βŸ©):(βŸ¨π‘“, π‘’βŸ©(Hom β€˜(𝑄 Γ—c 𝐢))βŸ¨π‘”, π‘£βŸ©)⟢((𝑓(1st β€˜πΈ)𝑒)(Hom β€˜π·)(𝑔(1st β€˜πΈ)𝑣))))
116107, 115bitrid 283 . . . . . . . . 9 (π‘₯ = βŸ¨π‘“, π‘’βŸ© β†’ (βˆ€π‘¦ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))(π‘₯(2nd β€˜πΈ)𝑦):(π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦)⟢(((1st β€˜πΈ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΈ)β€˜π‘¦)) ↔ βˆ€π‘” ∈ (𝐢 Func 𝐷)βˆ€π‘£ ∈ (Baseβ€˜πΆ)(βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘”, π‘£βŸ©):(βŸ¨π‘“, π‘’βŸ©(Hom β€˜(𝑄 Γ—c 𝐢))βŸ¨π‘”, π‘£βŸ©)⟢((𝑓(1st β€˜πΈ)𝑒)(Hom β€˜π·)(𝑔(1st β€˜πΈ)𝑣))))
117116ralxp 5842 . . . . . . . 8 (βˆ€π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))βˆ€π‘¦ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))(π‘₯(2nd β€˜πΈ)𝑦):(π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦)⟢(((1st β€˜πΈ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΈ)β€˜π‘¦)) ↔ βˆ€π‘“ ∈ (𝐢 Func 𝐷)βˆ€π‘’ ∈ (Baseβ€˜πΆ)βˆ€π‘” ∈ (𝐢 Func 𝐷)βˆ€π‘£ ∈ (Baseβ€˜πΆ)(βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘”, π‘£βŸ©):(βŸ¨π‘“, π‘’βŸ©(Hom β€˜(𝑄 Γ—c 𝐢))βŸ¨π‘”, π‘£βŸ©)⟢((𝑓(1st β€˜πΈ)𝑒)(Hom β€˜π·)(𝑔(1st β€˜πΈ)𝑣)))
11899, 117sylibr 233 . . . . . . 7 (πœ‘ β†’ βˆ€π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))βˆ€π‘¦ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))(π‘₯(2nd β€˜πΈ)𝑦):(π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦)⟢(((1st β€˜πΈ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΈ)β€˜π‘¦)))
119118r19.21bi 3249 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) β†’ βˆ€π‘¦ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))(π‘₯(2nd β€˜πΈ)𝑦):(π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦)⟢(((1st β€˜πΈ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΈ)β€˜π‘¦)))
120119r19.21bi 3249 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) β†’ (π‘₯(2nd β€˜πΈ)𝑦):(π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦)⟢(((1st β€˜πΈ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΈ)β€˜π‘¦)))
121120anasss 468 . . . 4 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)))) β†’ (π‘₯(2nd β€˜πΈ)𝑦):(π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦)⟢(((1st β€˜πΈ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΈ)β€˜π‘¦)))
12228adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ 𝑄 ∈ Cat)
1232adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ 𝐢 ∈ Cat)
124 eqid 2733 . . . . . . . . . . 11 (Idβ€˜π‘„) = (Idβ€˜π‘„)
125 eqid 2733 . . . . . . . . . . 11 (Idβ€˜πΆ) = (Idβ€˜πΆ)
126 simprl 770 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ 𝑓 ∈ (𝐢 Func 𝐷))
127 simprr 772 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ 𝑒 ∈ (Baseβ€˜πΆ))
12818, 122, 123, 20, 4, 124, 125, 25, 126, 127xpcid 18141 . . . . . . . . . 10 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ ((Idβ€˜(𝑄 Γ—c 𝐢))β€˜βŸ¨π‘“, π‘’βŸ©) = ⟨((Idβ€˜π‘„)β€˜π‘“), ((Idβ€˜πΆ)β€˜π‘’)⟩)
129128fveq2d 6896 . . . . . . . . 9 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ ((βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘“, π‘’βŸ©)β€˜((Idβ€˜(𝑄 Γ—c 𝐢))β€˜βŸ¨π‘“, π‘’βŸ©)) = ((βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘“, π‘’βŸ©)β€˜βŸ¨((Idβ€˜π‘„)β€˜π‘“), ((Idβ€˜πΆ)β€˜π‘’)⟩))
130 df-ov 7412 . . . . . . . . 9 (((Idβ€˜π‘„)β€˜π‘“)(βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘“, π‘’βŸ©)((Idβ€˜πΆ)β€˜π‘’)) = ((βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘“, π‘’βŸ©)β€˜βŸ¨((Idβ€˜π‘„)β€˜π‘“), ((Idβ€˜πΆ)β€˜π‘’)⟩)
131129, 130eqtr4di 2791 . . . . . . . 8 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ ((βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘“, π‘’βŸ©)β€˜((Idβ€˜(𝑄 Γ—c 𝐢))β€˜βŸ¨π‘“, π‘’βŸ©)) = (((Idβ€˜π‘„)β€˜π‘“)(βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘“, π‘’βŸ©)((Idβ€˜πΆ)β€˜π‘’)))
1323adantr 482 . . . . . . . . 9 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ 𝐷 ∈ Cat)
133 eqid 2733 . . . . . . . . 9 (βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘“, π‘’βŸ©) = (βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘“, π‘’βŸ©)
13420, 91, 124, 122, 126catidcl 17626 . . . . . . . . 9 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ ((Idβ€˜π‘„)β€˜π‘“) ∈ (𝑓(𝐢 Nat 𝐷)𝑓))
1354, 5, 125, 123, 127catidcl 17626 . . . . . . . . 9 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ ((Idβ€˜πΆ)β€˜π‘’) ∈ (𝑒(Hom β€˜πΆ)𝑒))
1361, 123, 132, 4, 5, 6, 7, 126, 126, 127, 127, 133, 134, 135evlf2val 18172 . . . . . . . 8 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ (((Idβ€˜π‘„)β€˜π‘“)(βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘“, π‘’βŸ©)((Idβ€˜πΆ)β€˜π‘’)) = ((((Idβ€˜π‘„)β€˜π‘“)β€˜π‘’)(⟨((1st β€˜π‘“)β€˜π‘’), ((1st β€˜π‘“)β€˜π‘’)⟩(compβ€˜π·)((1st β€˜π‘“)β€˜π‘’))((𝑒(2nd β€˜π‘“)𝑒)β€˜((Idβ€˜πΆ)β€˜π‘’))))
13730, 126, 32sylancr 588 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ (1st β€˜π‘“)(𝐢 Func 𝐷)(2nd β€˜π‘“))
1384, 22, 137funcf1 17816 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ (1st β€˜π‘“):(Baseβ€˜πΆ)⟢(Baseβ€˜π·))
139138, 127ffvelcdmd 7088 . . . . . . . . . 10 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ ((1st β€˜π‘“)β€˜π‘’) ∈ (Baseβ€˜π·))
14022, 24, 26, 132, 139catidcl 17626 . . . . . . . . . 10 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ ((Idβ€˜π·)β€˜((1st β€˜π‘“)β€˜π‘’)) ∈ (((1st β€˜π‘“)β€˜π‘’)(Hom β€˜π·)((1st β€˜π‘“)β€˜π‘’)))
14122, 24, 26, 132, 139, 6, 139, 140catlid 17627 . . . . . . . . 9 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ (((Idβ€˜π·)β€˜((1st β€˜π‘“)β€˜π‘’))(⟨((1st β€˜π‘“)β€˜π‘’), ((1st β€˜π‘“)β€˜π‘’)⟩(compβ€˜π·)((1st β€˜π‘“)β€˜π‘’))((Idβ€˜π·)β€˜((1st β€˜π‘“)β€˜π‘’))) = ((Idβ€˜π·)β€˜((1st β€˜π‘“)β€˜π‘’)))
14219, 124, 26, 126fucid 17924 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ ((Idβ€˜π‘„)β€˜π‘“) = ((Idβ€˜π·) ∘ (1st β€˜π‘“)))
143142fveq1d 6894 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ (((Idβ€˜π‘„)β€˜π‘“)β€˜π‘’) = (((Idβ€˜π·) ∘ (1st β€˜π‘“))β€˜π‘’))
144 fvco3 6991 . . . . . . . . . . . 12 (((1st β€˜π‘“):(Baseβ€˜πΆ)⟢(Baseβ€˜π·) ∧ 𝑒 ∈ (Baseβ€˜πΆ)) β†’ (((Idβ€˜π·) ∘ (1st β€˜π‘“))β€˜π‘’) = ((Idβ€˜π·)β€˜((1st β€˜π‘“)β€˜π‘’)))
145138, 127, 144syl2anc 585 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ (((Idβ€˜π·) ∘ (1st β€˜π‘“))β€˜π‘’) = ((Idβ€˜π·)β€˜((1st β€˜π‘“)β€˜π‘’)))
146143, 145eqtrd 2773 . . . . . . . . . 10 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ (((Idβ€˜π‘„)β€˜π‘“)β€˜π‘’) = ((Idβ€˜π·)β€˜((1st β€˜π‘“)β€˜π‘’)))
1474, 125, 26, 137, 127funcid 17820 . . . . . . . . . 10 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ ((𝑒(2nd β€˜π‘“)𝑒)β€˜((Idβ€˜πΆ)β€˜π‘’)) = ((Idβ€˜π·)β€˜((1st β€˜π‘“)β€˜π‘’)))
148146, 147oveq12d 7427 . . . . . . . . 9 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ ((((Idβ€˜π‘„)β€˜π‘“)β€˜π‘’)(⟨((1st β€˜π‘“)β€˜π‘’), ((1st β€˜π‘“)β€˜π‘’)⟩(compβ€˜π·)((1st β€˜π‘“)β€˜π‘’))((𝑒(2nd β€˜π‘“)𝑒)β€˜((Idβ€˜πΆ)β€˜π‘’))) = (((Idβ€˜π·)β€˜((1st β€˜π‘“)β€˜π‘’))(⟨((1st β€˜π‘“)β€˜π‘’), ((1st β€˜π‘“)β€˜π‘’)⟩(compβ€˜π·)((1st β€˜π‘“)β€˜π‘’))((Idβ€˜π·)β€˜((1st β€˜π‘“)β€˜π‘’))))
1491, 123, 132, 4, 126, 127evlf1 18173 . . . . . . . . . 10 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ (𝑓(1st β€˜πΈ)𝑒) = ((1st β€˜π‘“)β€˜π‘’))
150149fveq2d 6896 . . . . . . . . 9 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ ((Idβ€˜π·)β€˜(𝑓(1st β€˜πΈ)𝑒)) = ((Idβ€˜π·)β€˜((1st β€˜π‘“)β€˜π‘’)))
151141, 148, 1503eqtr4d 2783 . . . . . . . 8 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ ((((Idβ€˜π‘„)β€˜π‘“)β€˜π‘’)(⟨((1st β€˜π‘“)β€˜π‘’), ((1st β€˜π‘“)β€˜π‘’)⟩(compβ€˜π·)((1st β€˜π‘“)β€˜π‘’))((𝑒(2nd β€˜π‘“)𝑒)β€˜((Idβ€˜πΆ)β€˜π‘’))) = ((Idβ€˜π·)β€˜(𝑓(1st β€˜πΈ)𝑒)))
152131, 136, 1513eqtrd 2777 . . . . . . 7 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ ((βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘“, π‘’βŸ©)β€˜((Idβ€˜(𝑄 Γ—c 𝐢))β€˜βŸ¨π‘“, π‘’βŸ©)) = ((Idβ€˜π·)β€˜(𝑓(1st β€˜πΈ)𝑒)))
153152ralrimivva 3201 . . . . . 6 (πœ‘ β†’ βˆ€π‘“ ∈ (𝐢 Func 𝐷)βˆ€π‘’ ∈ (Baseβ€˜πΆ)((βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘“, π‘’βŸ©)β€˜((Idβ€˜(𝑄 Γ—c 𝐢))β€˜βŸ¨π‘“, π‘’βŸ©)) = ((Idβ€˜π·)β€˜(𝑓(1st β€˜πΈ)𝑒)))
154 id 22 . . . . . . . . . 10 (π‘₯ = βŸ¨π‘“, π‘’βŸ© β†’ π‘₯ = βŸ¨π‘“, π‘’βŸ©)
155154, 154oveq12d 7427 . . . . . . . . 9 (π‘₯ = βŸ¨π‘“, π‘’βŸ© β†’ (π‘₯(2nd β€˜πΈ)π‘₯) = (βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘“, π‘’βŸ©))
156 fveq2 6892 . . . . . . . . 9 (π‘₯ = βŸ¨π‘“, π‘’βŸ© β†’ ((Idβ€˜(𝑄 Γ—c 𝐢))β€˜π‘₯) = ((Idβ€˜(𝑄 Γ—c 𝐢))β€˜βŸ¨π‘“, π‘’βŸ©))
157155, 156fveq12d 6899 . . . . . . . 8 (π‘₯ = βŸ¨π‘“, π‘’βŸ© β†’ ((π‘₯(2nd β€˜πΈ)π‘₯)β€˜((Idβ€˜(𝑄 Γ—c 𝐢))β€˜π‘₯)) = ((βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘“, π‘’βŸ©)β€˜((Idβ€˜(𝑄 Γ—c 𝐢))β€˜βŸ¨π‘“, π‘’βŸ©)))
158112fveq2d 6896 . . . . . . . 8 (π‘₯ = βŸ¨π‘“, π‘’βŸ© β†’ ((Idβ€˜π·)β€˜((1st β€˜πΈ)β€˜π‘₯)) = ((Idβ€˜π·)β€˜(𝑓(1st β€˜πΈ)𝑒)))
159157, 158eqeq12d 2749 . . . . . . 7 (π‘₯ = βŸ¨π‘“, π‘’βŸ© β†’ (((π‘₯(2nd β€˜πΈ)π‘₯)β€˜((Idβ€˜(𝑄 Γ—c 𝐢))β€˜π‘₯)) = ((Idβ€˜π·)β€˜((1st β€˜πΈ)β€˜π‘₯)) ↔ ((βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘“, π‘’βŸ©)β€˜((Idβ€˜(𝑄 Γ—c 𝐢))β€˜βŸ¨π‘“, π‘’βŸ©)) = ((Idβ€˜π·)β€˜(𝑓(1st β€˜πΈ)𝑒))))
160159ralxp 5842 . . . . . 6 (βˆ€π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))((π‘₯(2nd β€˜πΈ)π‘₯)β€˜((Idβ€˜(𝑄 Γ—c 𝐢))β€˜π‘₯)) = ((Idβ€˜π·)β€˜((1st β€˜πΈ)β€˜π‘₯)) ↔ βˆ€π‘“ ∈ (𝐢 Func 𝐷)βˆ€π‘’ ∈ (Baseβ€˜πΆ)((βŸ¨π‘“, π‘’βŸ©(2nd β€˜πΈ)βŸ¨π‘“, π‘’βŸ©)β€˜((Idβ€˜(𝑄 Γ—c 𝐢))β€˜βŸ¨π‘“, π‘’βŸ©)) = ((Idβ€˜π·)β€˜(𝑓(1st β€˜πΈ)𝑒)))
161153, 160sylibr 233 . . . . 5 (πœ‘ β†’ βˆ€π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))((π‘₯(2nd β€˜πΈ)π‘₯)β€˜((Idβ€˜(𝑄 Γ—c 𝐢))β€˜π‘₯)) = ((Idβ€˜π·)β€˜((1st β€˜πΈ)β€˜π‘₯)))
162161r19.21bi 3249 . . . 4 ((πœ‘ ∧ π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) β†’ ((π‘₯(2nd β€˜πΈ)π‘₯)β€˜((Idβ€˜(𝑄 Γ—c 𝐢))β€˜π‘₯)) = ((Idβ€˜π·)β€˜((1st β€˜πΈ)β€˜π‘₯)))
16323ad2ant1 1134 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ 𝐢 ∈ Cat)
16433ad2ant1 1134 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ 𝐷 ∈ Cat)
165 simp21 1207 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)))
166 1st2nd2 8014 . . . . . . . . 9 (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) β†’ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩)
167165, 166syl 17 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩)
168167, 165eqeltrrd 2835 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)))
169 opelxp 5713 . . . . . . 7 (⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ↔ ((1st β€˜π‘₯) ∈ (𝐢 Func 𝐷) ∧ (2nd β€˜π‘₯) ∈ (Baseβ€˜πΆ)))
170168, 169sylib 217 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ ((1st β€˜π‘₯) ∈ (𝐢 Func 𝐷) ∧ (2nd β€˜π‘₯) ∈ (Baseβ€˜πΆ)))
171 simp22 1208 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)))
172 1st2nd2 8014 . . . . . . . . 9 (𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) β†’ 𝑦 = ⟨(1st β€˜π‘¦), (2nd β€˜π‘¦)⟩)
173171, 172syl 17 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ 𝑦 = ⟨(1st β€˜π‘¦), (2nd β€˜π‘¦)⟩)
174173, 171eqeltrrd 2835 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ ⟨(1st β€˜π‘¦), (2nd β€˜π‘¦)⟩ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)))
175 opelxp 5713 . . . . . . 7 (⟨(1st β€˜π‘¦), (2nd β€˜π‘¦)⟩ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ↔ ((1st β€˜π‘¦) ∈ (𝐢 Func 𝐷) ∧ (2nd β€˜π‘¦) ∈ (Baseβ€˜πΆ)))
176174, 175sylib 217 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ ((1st β€˜π‘¦) ∈ (𝐢 Func 𝐷) ∧ (2nd β€˜π‘¦) ∈ (Baseβ€˜πΆ)))
177 simp23 1209 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)))
178 1st2nd2 8014 . . . . . . . . 9 (𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) β†’ 𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
179177, 178syl 17 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ 𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
180179, 177eqeltrrd 2835 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)))
181 opelxp 5713 . . . . . . 7 (⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ↔ ((1st β€˜π‘§) ∈ (𝐢 Func 𝐷) ∧ (2nd β€˜π‘§) ∈ (Baseβ€˜πΆ)))
182180, 181sylib 217 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ ((1st β€˜π‘§) ∈ (𝐢 Func 𝐷) ∧ (2nd β€˜π‘§) ∈ (Baseβ€˜πΆ)))
183 simp3l 1202 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ 𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦))
18418, 21, 91, 5, 23, 165, 171xpchom 18132 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) = (((1st β€˜π‘₯)(𝐢 Nat 𝐷)(1st β€˜π‘¦)) Γ— ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦))))
185183, 184eleqtrd 2836 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ 𝑓 ∈ (((1st β€˜π‘₯)(𝐢 Nat 𝐷)(1st β€˜π‘¦)) Γ— ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦))))
186 1st2nd2 8014 . . . . . . . . 9 (𝑓 ∈ (((1st β€˜π‘₯)(𝐢 Nat 𝐷)(1st β€˜π‘¦)) Γ— ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦))) β†’ 𝑓 = ⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩)
187185, 186syl 17 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ 𝑓 = ⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩)
188187, 185eqeltrrd 2835 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ ⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩ ∈ (((1st β€˜π‘₯)(𝐢 Nat 𝐷)(1st β€˜π‘¦)) Γ— ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦))))
189 opelxp 5713 . . . . . . 7 (⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩ ∈ (((1st β€˜π‘₯)(𝐢 Nat 𝐷)(1st β€˜π‘¦)) Γ— ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦))) ↔ ((1st β€˜π‘“) ∈ ((1st β€˜π‘₯)(𝐢 Nat 𝐷)(1st β€˜π‘¦)) ∧ (2nd β€˜π‘“) ∈ ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦))))
190188, 189sylib 217 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ ((1st β€˜π‘“) ∈ ((1st β€˜π‘₯)(𝐢 Nat 𝐷)(1st β€˜π‘¦)) ∧ (2nd β€˜π‘“) ∈ ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦))))
191 simp3r 1203 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))
19218, 21, 91, 5, 23, 171, 177xpchom 18132 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧) = (((1st β€˜π‘¦)(𝐢 Nat 𝐷)(1st β€˜π‘§)) Γ— ((2nd β€˜π‘¦)(Hom β€˜πΆ)(2nd β€˜π‘§))))
193191, 192eleqtrd 2836 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ 𝑔 ∈ (((1st β€˜π‘¦)(𝐢 Nat 𝐷)(1st β€˜π‘§)) Γ— ((2nd β€˜π‘¦)(Hom β€˜πΆ)(2nd β€˜π‘§))))
194 1st2nd2 8014 . . . . . . . . 9 (𝑔 ∈ (((1st β€˜π‘¦)(𝐢 Nat 𝐷)(1st β€˜π‘§)) Γ— ((2nd β€˜π‘¦)(Hom β€˜πΆ)(2nd β€˜π‘§))) β†’ 𝑔 = ⟨(1st β€˜π‘”), (2nd β€˜π‘”)⟩)
195193, 194syl 17 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ 𝑔 = ⟨(1st β€˜π‘”), (2nd β€˜π‘”)⟩)
196195, 193eqeltrrd 2835 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ ⟨(1st β€˜π‘”), (2nd β€˜π‘”)⟩ ∈ (((1st β€˜π‘¦)(𝐢 Nat 𝐷)(1st β€˜π‘§)) Γ— ((2nd β€˜π‘¦)(Hom β€˜πΆ)(2nd β€˜π‘§))))
197 opelxp 5713 . . . . . . 7 (⟨(1st β€˜π‘”), (2nd β€˜π‘”)⟩ ∈ (((1st β€˜π‘¦)(𝐢 Nat 𝐷)(1st β€˜π‘§)) Γ— ((2nd β€˜π‘¦)(Hom β€˜πΆ)(2nd β€˜π‘§))) ↔ ((1st β€˜π‘”) ∈ ((1st β€˜π‘¦)(𝐢 Nat 𝐷)(1st β€˜π‘§)) ∧ (2nd β€˜π‘”) ∈ ((2nd β€˜π‘¦)(Hom β€˜πΆ)(2nd β€˜π‘§))))
198196, 197sylib 217 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ ((1st β€˜π‘”) ∈ ((1st β€˜π‘¦)(𝐢 Nat 𝐷)(1st β€˜π‘§)) ∧ (2nd β€˜π‘”) ∈ ((2nd β€˜π‘¦)(Hom β€˜πΆ)(2nd β€˜π‘§))))
1991, 19, 163, 164, 7, 170, 176, 182, 190, 198evlfcllem 18174 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ ((⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)β€˜(⟨(1st β€˜π‘”), (2nd β€˜π‘”)⟩(⟨⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩, ⟨(1st β€˜π‘¦), (2nd β€˜π‘¦)⟩⟩(compβ€˜(𝑄 Γ—c 𝐢))⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩)) = (((⟨(1st β€˜π‘¦), (2nd β€˜π‘¦)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)β€˜βŸ¨(1st β€˜π‘”), (2nd β€˜π‘”)⟩)(⟨((1st β€˜πΈ)β€˜βŸ¨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩), ((1st β€˜πΈ)β€˜βŸ¨(1st β€˜π‘¦), (2nd β€˜π‘¦)⟩)⟩(compβ€˜π·)((1st β€˜πΈ)β€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩))((⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘¦), (2nd β€˜π‘¦)⟩)β€˜βŸ¨(1st β€˜π‘“), (2nd β€˜π‘“)⟩)))
200167, 179oveq12d 7427 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ (π‘₯(2nd β€˜πΈ)𝑧) = (⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩))
201167, 173opeq12d 4882 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ ⟨π‘₯, π‘¦βŸ© = ⟨⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩, ⟨(1st β€˜π‘¦), (2nd β€˜π‘¦)⟩⟩)
202201, 179oveq12d 7427 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ (⟨π‘₯, π‘¦βŸ©(compβ€˜(𝑄 Γ—c 𝐢))𝑧) = (⟨⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩, ⟨(1st β€˜π‘¦), (2nd β€˜π‘¦)⟩⟩(compβ€˜(𝑄 Γ—c 𝐢))⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩))
203202, 195, 187oveq123d 7430 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ (𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜(𝑄 Γ—c 𝐢))𝑧)𝑓) = (⟨(1st β€˜π‘”), (2nd β€˜π‘”)⟩(⟨⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩, ⟨(1st β€˜π‘¦), (2nd β€˜π‘¦)⟩⟩(compβ€˜(𝑄 Γ—c 𝐢))⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩))
204200, 203fveq12d 6899 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ ((π‘₯(2nd β€˜πΈ)𝑧)β€˜(𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜(𝑄 Γ—c 𝐢))𝑧)𝑓)) = ((⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)β€˜(⟨(1st β€˜π‘”), (2nd β€˜π‘”)⟩(⟨⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩, ⟨(1st β€˜π‘¦), (2nd β€˜π‘¦)⟩⟩(compβ€˜(𝑄 Γ—c 𝐢))⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)⟨(1st β€˜π‘“), (2nd β€˜π‘“)⟩)))
205167fveq2d 6896 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ ((1st β€˜πΈ)β€˜π‘₯) = ((1st β€˜πΈ)β€˜βŸ¨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩))
206173fveq2d 6896 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ ((1st β€˜πΈ)β€˜π‘¦) = ((1st β€˜πΈ)β€˜βŸ¨(1st β€˜π‘¦), (2nd β€˜π‘¦)⟩))
207205, 206opeq12d 4882 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ ⟨((1st β€˜πΈ)β€˜π‘₯), ((1st β€˜πΈ)β€˜π‘¦)⟩ = ⟨((1st β€˜πΈ)β€˜βŸ¨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩), ((1st β€˜πΈ)β€˜βŸ¨(1st β€˜π‘¦), (2nd β€˜π‘¦)⟩)⟩)
208179fveq2d 6896 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ ((1st β€˜πΈ)β€˜π‘§) = ((1st β€˜πΈ)β€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩))
209207, 208oveq12d 7427 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ (⟨((1st β€˜πΈ)β€˜π‘₯), ((1st β€˜πΈ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΈ)β€˜π‘§)) = (⟨((1st β€˜πΈ)β€˜βŸ¨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩), ((1st β€˜πΈ)β€˜βŸ¨(1st β€˜π‘¦), (2nd β€˜π‘¦)⟩)⟩(compβ€˜π·)((1st β€˜πΈ)β€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)))
210173, 179oveq12d 7427 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ (𝑦(2nd β€˜πΈ)𝑧) = (⟨(1st β€˜π‘¦), (2nd β€˜π‘¦)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩))
211210, 195fveq12d 6899 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ ((𝑦(2nd β€˜πΈ)𝑧)β€˜π‘”) = ((⟨(1st β€˜π‘¦), (2nd β€˜π‘¦)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)β€˜βŸ¨(1st β€˜π‘”), (2nd β€˜π‘”)⟩))
212167, 173oveq12d 7427 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ (π‘₯(2nd β€˜πΈ)𝑦) = (⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘¦), (2nd β€˜π‘¦)⟩))
213212, 187fveq12d 6899 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ ((π‘₯(2nd β€˜πΈ)𝑦)β€˜π‘“) = ((⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘¦), (2nd β€˜π‘¦)⟩)β€˜βŸ¨(1st β€˜π‘“), (2nd β€˜π‘“)⟩))
214209, 211, 213oveq123d 7430 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ (((𝑦(2nd β€˜πΈ)𝑧)β€˜π‘”)(⟨((1st β€˜πΈ)β€˜π‘₯), ((1st β€˜πΈ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΈ)β€˜π‘§))((π‘₯(2nd β€˜πΈ)𝑦)β€˜π‘“)) = (((⟨(1st β€˜π‘¦), (2nd β€˜π‘¦)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)β€˜βŸ¨(1st β€˜π‘”), (2nd β€˜π‘”)⟩)(⟨((1st β€˜πΈ)β€˜βŸ¨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩), ((1st β€˜πΈ)β€˜βŸ¨(1st β€˜π‘¦), (2nd β€˜π‘¦)⟩)⟩(compβ€˜π·)((1st β€˜πΈ)β€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩))((⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘¦), (2nd β€˜π‘¦)⟩)β€˜βŸ¨(1st β€˜π‘“), (2nd β€˜π‘“)⟩)))
215199, 204, 2143eqtr4d 2783 . . . 4 ((πœ‘ ∧ (π‘₯ ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑦 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∧ 𝑧 ∈ ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜(𝑄 Γ—c 𝐢))𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜(𝑄 Γ—c 𝐢))𝑧))) β†’ ((π‘₯(2nd β€˜πΈ)𝑧)β€˜(𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜(𝑄 Γ—c 𝐢))𝑧)𝑓)) = (((𝑦(2nd β€˜πΈ)𝑧)β€˜π‘”)(⟨((1st β€˜πΈ)β€˜π‘₯), ((1st β€˜πΈ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΈ)β€˜π‘§))((π‘₯(2nd β€˜πΈ)𝑦)β€˜π‘“)))
21621, 22, 23, 24, 25, 26, 27, 6, 29, 3, 44, 55, 121, 162, 215isfuncd 17815 . . 3 (πœ‘ β†’ (1st β€˜πΈ)((𝑄 Γ—c 𝐢) Func 𝐷)(2nd β€˜πΈ))
217 df-br 5150 . . 3 ((1st β€˜πΈ)((𝑄 Γ—c 𝐢) Func 𝐷)(2nd β€˜πΈ) ↔ ⟨(1st β€˜πΈ), (2nd β€˜πΈ)⟩ ∈ ((𝑄 Γ—c 𝐢) Func 𝐷))
218216, 217sylib 217 . 2 (πœ‘ β†’ ⟨(1st β€˜πΈ), (2nd β€˜πΈ)⟩ ∈ ((𝑄 Γ—c 𝐢) Func 𝐷))
21917, 218eqeltrd 2834 1 (πœ‘ β†’ 𝐸 ∈ ((𝑄 Γ—c 𝐢) Func 𝐷))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  Vcvv 3475  β¦‹csb 3894  βŸ¨cop 4635   class class class wbr 5149   Γ— cxp 5675   ∘ ccom 5681  Rel wrel 5682   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  1st c1st 7973  2nd c2nd 7974  Basecbs 17144  Hom chom 17208  compcco 17209  Catccat 17608  Idccid 17609   Func cfunc 17804   Nat cnat 17892   FuncCat cfuc 17893   Γ—c cxpc 18120   evalF cevlf 18162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-fz 13485  df-struct 17080  df-slot 17115  df-ndx 17127  df-base 17145  df-hom 17221  df-cco 17222  df-cat 17612  df-cid 17613  df-func 17808  df-nat 17894  df-fuc 17895  df-xpc 18124  df-evlf 18166
This theorem is referenced by:  uncfcl  18188  uncf1  18189  uncf2  18190  yonedalem1  18225
  Copyright terms: Public domain W3C validator