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Theorem evlfcl 18116
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 18111 . . . 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 7391 . . . . . 6 (𝐢 Func 𝐷) ∈ V
10 fvex 6856 . . . . . 6 (Baseβ€˜πΆ) ∈ V
119, 10mpoex 8013 . . . . 5 (𝑓 ∈ (𝐢 Func 𝐷), π‘₯ ∈ (Baseβ€˜πΆ) ↦ ((1st β€˜π‘“)β€˜π‘₯)) ∈ V
129, 10xpex 7688 . . . . . 6 ((𝐢 Func 𝐷) Γ— (Baseβ€˜πΆ)) ∈ V
1312, 12mpoex 8013 . . . . 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 5673 . . . 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 7961 . . 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 17853 . . . . 5 (𝐢 Func 𝐷) = (Baseβ€˜π‘„)
2118, 20, 4xpcbas 18071 . . . 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 17864 . . . . 5 (πœ‘ β†’ 𝑄 ∈ Cat)
2918, 28, 2xpccat 18083 . . . 4 (πœ‘ β†’ (𝑄 Γ—c 𝐢) ∈ Cat)
30 relfunc 17753 . . . . . . . . . . 11 Rel (𝐢 Func 𝐷)
31 simpr 486 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑓 ∈ (𝐢 Func 𝐷)) β†’ 𝑓 ∈ (𝐢 Func 𝐷))
32 1st2ndbr 7975 . . . . . . . . . . 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 17757 . . . . . . . . 9 ((πœ‘ ∧ 𝑓 ∈ (𝐢 Func 𝐷)) β†’ (1st β€˜π‘“):(Baseβ€˜πΆ)⟢(Baseβ€˜π·))
3534ffvelcdmda 7036 . . . . . . . 8 (((πœ‘ ∧ 𝑓 ∈ (𝐢 Func 𝐷)) ∧ π‘₯ ∈ (Baseβ€˜πΆ)) β†’ ((1st β€˜π‘“)β€˜π‘₯) ∈ (Baseβ€˜π·))
3635ralrimiva 3140 . . . . . . 7 ((πœ‘ ∧ 𝑓 ∈ (𝐢 Func 𝐷)) β†’ βˆ€π‘₯ ∈ (Baseβ€˜πΆ)((1st β€˜π‘“)β€˜π‘₯) ∈ (Baseβ€˜π·))
3736ralrimiva 3140 . . . . . 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 8001 . . . . . 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 7932 . . . . . . 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 6654 . . . . 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 7391 . . . . . . . . 9 (π‘š(𝐢 Nat 𝐷)𝑛) ∈ V
47 ovex 7391 . . . . . . . . 9 ((2nd β€˜π‘₯)(Hom β€˜πΆ)(2nd β€˜π‘¦)) ∈ V
4846, 47mpoex 8013 . . . . . . . 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 5269 . . . . . . 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 5269 . . . . . 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 8003 . . . . 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 7933 . . . . . . 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 6596 . . . . 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 17757 . . . . . . . . . . . . . . . . 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 7037 . . . . . . . . . . . . . . 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 7037 . . . . . . . . . . . . . . 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 7975 . . . . . . . . . . . . . . . . . . 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 17757 . . . . . . . . . . . . . . . . 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 7037 . . . . . . . . . . . . . . 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 17759 . . . . . . . . . . . . . . . . 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 7037 . . . . . . . . . . . . . . 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 17843 . . . . . . . . . . . . . . . 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 17845 . . . . . . . . . . . . . . 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 17570 . . . . . . . . . . . . . 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 3194 . . . . . . . . . . . . 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 8001 . . . . . . . . . . . . 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 18112 . . . . . . . . . . . . 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 6654 . . . . . . . . . . . 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 17854 . . . . . . . . . . . . 13 (𝐢 Nat 𝐷) = (Hom β€˜π‘„)
9218, 20, 4, 91, 5, 58, 62, 67, 73, 23xpchom2 18079 . . . . . . . . . . . 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 18114 . . . . . . . . . . . . 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 18114 . . . . . . . . . . . . 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 7376 . . . . . . . . . . . 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 6664 . . . . . . . . . . 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 3194 . . . . . . . . 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 3194 . . . . . . . 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 7366 . . . . . . . . . . . 12 (𝑦 = βŸ¨π‘”, π‘£βŸ© β†’ (π‘₯(2nd β€˜πΈ)𝑦) = (π‘₯(2nd β€˜πΈ)βŸ¨π‘”, π‘£βŸ©))
101 oveq2 7366 . . . . . . . . . . . 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 6843 . . . . . . . . . . . . . 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 7361 . . . . . . . . . . . . . 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 7374 . . . . . . . . . . . 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 6658 . . . . . . . . . . 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 5798 . . . . . . . . . 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 7365 . . . . . . . . . . . 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 7365 . . . . . . . . . . . 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 6843 . . . . . . . . . . . . . 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 7361 . . . . . . . . . . . . . 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 7373 . . . . . . . . . . . 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 6658 . . . . . . . . . . 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 3209 . . . . . . . . . 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 5798 . . . . . . . 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 3233 . . . . . 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 3233 . . . . 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 18082 . . . . . . . . . 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 6847 . . . . . . . . 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 7361 . . . . . . . . 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 17567 . . . . . . . . 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 17567 . . . . . . . . 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 18113 . . . . . . . 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 17757 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ (1st β€˜π‘“):(Baseβ€˜πΆ)⟢(Baseβ€˜π·))
139138, 127ffvelcdmd 7037 . . . . . . . . . 10 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ ((1st β€˜π‘“)β€˜π‘’) ∈ (Baseβ€˜π·))
14022, 24, 26, 132, 139catidcl 17567 . . . . . . . . . 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 17568 . . . . . . . . 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 17865 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ ((Idβ€˜π‘„)β€˜π‘“) = ((Idβ€˜π·) ∘ (1st β€˜π‘“)))
143142fveq1d 6845 . . . . . . . . . . 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 6941 . . . . . . . . . . . 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 17761 . . . . . . . . . 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 7376 . . . . . . . . 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 18114 . . . . . . . . . 10 ((πœ‘ ∧ (𝑓 ∈ (𝐢 Func 𝐷) ∧ 𝑒 ∈ (Baseβ€˜πΆ))) β†’ (𝑓(1st β€˜πΈ)𝑒) = ((1st β€˜π‘“)β€˜π‘’))
150149fveq2d 6847 . . . . . . . . 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 3194 . . . . . 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 7376 . . . . . . . . 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 6843 . . . . . . . . 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 6850 . . . . . . . 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 6847 . . . . . . . 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 5798 . . . . . 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 3233 . . . 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 7961 . . . . . . . . 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 5670 . . . . . . 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 7961 . . . . . . . . 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 5670 . . . . . . 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 7961 . . . . . . . . 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 5670 . . . . . . 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 18073 . . . . . . . . . 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 7961 . . . . . . . . 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 5670 . . . . . . 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 18073 . . . . . . . . . 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 7961 . . . . . . . . 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 5670 . . . . . . 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 18115 . . . . 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 7376 . . . . . 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 4839 . . . . . . . 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 7376 . . . . . . 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 7379 . . . . . 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 6850 . . . . 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 6847 . . . . . . . 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 6847 . . . . . . . 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 4839 . . . . . . 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 6847 . . . . . . 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 7376 . . . . . 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 7376 . . . . . . 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 6850 . . . . . 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 7376 . . . . . . 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 6850 . . . . . 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 7379 . . . . 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 17756 . . 3 (πœ‘ β†’ (1st β€˜πΈ)((𝑄 Γ—c 𝐢) Func 𝐷)(2nd β€˜πΈ))
217 df-br 5107 . . 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 3061  Vcvv 3444  β¦‹csb 3856  βŸ¨cop 4593   class class class wbr 5106   Γ— cxp 5632   ∘ ccom 5638  Rel wrel 5639   Fn wfn 6492  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  1st c1st 7920  2nd c2nd 7921  Basecbs 17088  Hom chom 17149  compcco 17150  Catccat 17549  Idccid 17550   Func cfunc 17745   Nat cnat 17833   FuncCat cfuc 17834   Γ—c cxpc 18061   evalF cevlf 18103
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-map 8770  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-fz 13431  df-struct 17024  df-slot 17059  df-ndx 17071  df-base 17089  df-hom 17162  df-cco 17163  df-cat 17553  df-cid 17554  df-func 17749  df-nat 17835  df-fuc 17836  df-xpc 18065  df-evlf 18107
This theorem is referenced by:  uncfcl  18129  uncf1  18130  uncf2  18131  yonedalem1  18166
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