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Theorem yonedalem3 18177
Description: Lemma for yoneda 18180. (Contributed by Mario Carneiro, 28-Jan-2017.)
Hypotheses
Ref Expression
yoneda.y π‘Œ = (Yonβ€˜πΆ)
yoneda.b 𝐡 = (Baseβ€˜πΆ)
yoneda.1 1 = (Idβ€˜πΆ)
yoneda.o 𝑂 = (oppCatβ€˜πΆ)
yoneda.s 𝑆 = (SetCatβ€˜π‘ˆ)
yoneda.t 𝑇 = (SetCatβ€˜π‘‰)
yoneda.q 𝑄 = (𝑂 FuncCat 𝑆)
yoneda.h 𝐻 = (HomFβ€˜π‘„)
yoneda.r 𝑅 = ((𝑄 Γ—c 𝑂) FuncCat 𝑇)
yoneda.e 𝐸 = (𝑂 evalF 𝑆)
yoneda.z 𝑍 = (𝐻 ∘func ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
yoneda.c (πœ‘ β†’ 𝐢 ∈ Cat)
yoneda.w (πœ‘ β†’ 𝑉 ∈ π‘Š)
yoneda.u (πœ‘ β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
yoneda.v (πœ‘ β†’ (ran (Homf β€˜π‘„) βˆͺ π‘ˆ) βŠ† 𝑉)
yoneda.m 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), π‘₯ ∈ 𝐡 ↦ (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘₯)(𝑂 Nat 𝑆)𝑓) ↦ ((π‘Žβ€˜π‘₯)β€˜( 1 β€˜π‘₯))))
Assertion
Ref Expression
yonedalem3 (πœ‘ β†’ 𝑀 ∈ (𝑍((𝑄 Γ—c 𝑂) Nat 𝑇)𝐸))
Distinct variable groups:   𝑓,π‘Ž,π‘₯, 1   𝐢,π‘Ž,𝑓,π‘₯   𝐸,π‘Ž,𝑓   𝐡,π‘Ž,𝑓,π‘₯   𝑂,π‘Ž,𝑓,π‘₯   𝑆,π‘Ž,𝑓,π‘₯   𝑄,π‘Ž,𝑓,π‘₯   𝑇,𝑓   πœ‘,π‘Ž,𝑓,π‘₯   π‘Œ,π‘Ž,𝑓,π‘₯   𝑍,π‘Ž,𝑓,π‘₯
Allowed substitution hints:   𝑅(π‘₯,𝑓,π‘Ž)   𝑇(π‘₯,π‘Ž)   π‘ˆ(π‘₯,𝑓,π‘Ž)   𝐸(π‘₯)   𝐻(π‘₯,𝑓,π‘Ž)   𝑀(π‘₯,𝑓,π‘Ž)   𝑉(π‘₯,𝑓,π‘Ž)   π‘Š(π‘₯,𝑓,π‘Ž)

Proof of Theorem yonedalem3
Dummy variables 𝑔 𝑦 𝑀 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 yoneda.m . . . . 5 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), π‘₯ ∈ 𝐡 ↦ (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘₯)(𝑂 Nat 𝑆)𝑓) ↦ ((π‘Žβ€˜π‘₯)β€˜( 1 β€˜π‘₯))))
2 ovex 7394 . . . . . 6 (((1st β€˜π‘Œ)β€˜π‘₯)(𝑂 Nat 𝑆)𝑓) ∈ V
32mptex 7177 . . . . 5 (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘₯)(𝑂 Nat 𝑆)𝑓) ↦ ((π‘Žβ€˜π‘₯)β€˜( 1 β€˜π‘₯))) ∈ V
41, 3fnmpoi 8006 . . . 4 𝑀 Fn ((𝑂 Func 𝑆) Γ— 𝐡)
54a1i 11 . . 3 (πœ‘ β†’ 𝑀 Fn ((𝑂 Func 𝑆) Γ— 𝐡))
6 yoneda.y . . . . . . . 8 π‘Œ = (Yonβ€˜πΆ)
7 yoneda.b . . . . . . . 8 𝐡 = (Baseβ€˜πΆ)
8 yoneda.1 . . . . . . . 8 1 = (Idβ€˜πΆ)
9 yoneda.o . . . . . . . 8 𝑂 = (oppCatβ€˜πΆ)
10 yoneda.s . . . . . . . 8 𝑆 = (SetCatβ€˜π‘ˆ)
11 yoneda.t . . . . . . . 8 𝑇 = (SetCatβ€˜π‘‰)
12 yoneda.q . . . . . . . 8 𝑄 = (𝑂 FuncCat 𝑆)
13 yoneda.h . . . . . . . 8 𝐻 = (HomFβ€˜π‘„)
14 yoneda.r . . . . . . . 8 𝑅 = ((𝑄 Γ—c 𝑂) FuncCat 𝑇)
15 yoneda.e . . . . . . . 8 𝐸 = (𝑂 evalF 𝑆)
16 yoneda.z . . . . . . . 8 𝑍 = (𝐻 ∘func ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
17 yoneda.c . . . . . . . . 9 (πœ‘ β†’ 𝐢 ∈ Cat)
1817adantr 482 . . . . . . . 8 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ 𝐢 ∈ Cat)
19 yoneda.w . . . . . . . . 9 (πœ‘ β†’ 𝑉 ∈ π‘Š)
2019adantr 482 . . . . . . . 8 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ 𝑉 ∈ π‘Š)
21 yoneda.u . . . . . . . . 9 (πœ‘ β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
2221adantr 482 . . . . . . . 8 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
23 yoneda.v . . . . . . . . 9 (πœ‘ β†’ (ran (Homf β€˜π‘„) βˆͺ π‘ˆ) βŠ† 𝑉)
2423adantr 482 . . . . . . . 8 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ (ran (Homf β€˜π‘„) βˆͺ π‘ˆ) βŠ† 𝑉)
25 simprl 770 . . . . . . . 8 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ 𝑔 ∈ (𝑂 Func 𝑆))
26 simprr 772 . . . . . . . 8 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ 𝑦 ∈ 𝐡)
276, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 25, 26, 1yonedalem3a 18171 . . . . . . 7 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ ((𝑔𝑀𝑦) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘¦)(𝑂 Nat 𝑆)𝑔) ↦ ((π‘Žβ€˜π‘¦)β€˜( 1 β€˜π‘¦))) ∧ (𝑔𝑀𝑦):(𝑔(1st β€˜π‘)𝑦)⟢(𝑔(1st β€˜πΈ)𝑦)))
2827simprd 497 . . . . . 6 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ (𝑔𝑀𝑦):(𝑔(1st β€˜π‘)𝑦)⟢(𝑔(1st β€˜πΈ)𝑦))
29 eqid 2733 . . . . . . 7 (Hom β€˜π‘‡) = (Hom β€˜π‘‡)
30 eqid 2733 . . . . . . . . . . 11 (𝑄 Γ—c 𝑂) = (𝑄 Γ—c 𝑂)
3112fucbas 17856 . . . . . . . . . . 11 (𝑂 Func 𝑆) = (Baseβ€˜π‘„)
329, 7oppcbas 17607 . . . . . . . . . . 11 𝐡 = (Baseβ€˜π‘‚)
3330, 31, 32xpcbas 18074 . . . . . . . . . 10 ((𝑂 Func 𝑆) Γ— 𝐡) = (Baseβ€˜(𝑄 Γ—c 𝑂))
34 eqid 2733 . . . . . . . . . 10 (Baseβ€˜π‘‡) = (Baseβ€˜π‘‡)
35 relfunc 17756 . . . . . . . . . . 11 Rel ((𝑄 Γ—c 𝑂) Func 𝑇)
366, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 23yonedalem1 18169 . . . . . . . . . . . 12 (πœ‘ β†’ (𝑍 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇)))
3736simpld 496 . . . . . . . . . . 11 (πœ‘ β†’ 𝑍 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇))
38 1st2ndbr 7978 . . . . . . . . . . 11 ((Rel ((𝑄 Γ—c 𝑂) Func 𝑇) ∧ 𝑍 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇)) β†’ (1st β€˜π‘)((𝑄 Γ—c 𝑂) Func 𝑇)(2nd β€˜π‘))
3935, 37, 38sylancr 588 . . . . . . . . . 10 (πœ‘ β†’ (1st β€˜π‘)((𝑄 Γ—c 𝑂) Func 𝑇)(2nd β€˜π‘))
4033, 34, 39funcf1 17760 . . . . . . . . 9 (πœ‘ β†’ (1st β€˜π‘):((𝑂 Func 𝑆) Γ— 𝐡)⟢(Baseβ€˜π‘‡))
4140fovcdmda 7529 . . . . . . . 8 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ (𝑔(1st β€˜π‘)𝑦) ∈ (Baseβ€˜π‘‡))
4211, 20setcbas 17972 . . . . . . . 8 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ 𝑉 = (Baseβ€˜π‘‡))
4341, 42eleqtrrd 2837 . . . . . . 7 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ (𝑔(1st β€˜π‘)𝑦) ∈ 𝑉)
4436simprd 497 . . . . . . . . . . 11 (πœ‘ β†’ 𝐸 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇))
45 1st2ndbr 7978 . . . . . . . . . . 11 ((Rel ((𝑄 Γ—c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇)) β†’ (1st β€˜πΈ)((𝑄 Γ—c 𝑂) Func 𝑇)(2nd β€˜πΈ))
4635, 44, 45sylancr 588 . . . . . . . . . 10 (πœ‘ β†’ (1st β€˜πΈ)((𝑄 Γ—c 𝑂) Func 𝑇)(2nd β€˜πΈ))
4733, 34, 46funcf1 17760 . . . . . . . . 9 (πœ‘ β†’ (1st β€˜πΈ):((𝑂 Func 𝑆) Γ— 𝐡)⟢(Baseβ€˜π‘‡))
4847fovcdmda 7529 . . . . . . . 8 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ (𝑔(1st β€˜πΈ)𝑦) ∈ (Baseβ€˜π‘‡))
4948, 42eleqtrrd 2837 . . . . . . 7 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ (𝑔(1st β€˜πΈ)𝑦) ∈ 𝑉)
5011, 20, 29, 43, 49elsetchom 17975 . . . . . 6 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ ((𝑔𝑀𝑦) ∈ ((𝑔(1st β€˜π‘)𝑦)(Hom β€˜π‘‡)(𝑔(1st β€˜πΈ)𝑦)) ↔ (𝑔𝑀𝑦):(𝑔(1st β€˜π‘)𝑦)⟢(𝑔(1st β€˜πΈ)𝑦)))
5128, 50mpbird 257 . . . . 5 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ (𝑔𝑀𝑦) ∈ ((𝑔(1st β€˜π‘)𝑦)(Hom β€˜π‘‡)(𝑔(1st β€˜πΈ)𝑦)))
5251ralrimivva 3194 . . . 4 (πœ‘ β†’ βˆ€π‘” ∈ (𝑂 Func 𝑆)βˆ€π‘¦ ∈ 𝐡 (𝑔𝑀𝑦) ∈ ((𝑔(1st β€˜π‘)𝑦)(Hom β€˜π‘‡)(𝑔(1st β€˜πΈ)𝑦)))
53 fveq2 6846 . . . . . . 7 (𝑧 = βŸ¨π‘”, π‘¦βŸ© β†’ (π‘€β€˜π‘§) = (π‘€β€˜βŸ¨π‘”, π‘¦βŸ©))
54 df-ov 7364 . . . . . . 7 (𝑔𝑀𝑦) = (π‘€β€˜βŸ¨π‘”, π‘¦βŸ©)
5553, 54eqtr4di 2791 . . . . . 6 (𝑧 = βŸ¨π‘”, π‘¦βŸ© β†’ (π‘€β€˜π‘§) = (𝑔𝑀𝑦))
56 fveq2 6846 . . . . . . . 8 (𝑧 = βŸ¨π‘”, π‘¦βŸ© β†’ ((1st β€˜π‘)β€˜π‘§) = ((1st β€˜π‘)β€˜βŸ¨π‘”, π‘¦βŸ©))
57 df-ov 7364 . . . . . . . 8 (𝑔(1st β€˜π‘)𝑦) = ((1st β€˜π‘)β€˜βŸ¨π‘”, π‘¦βŸ©)
5856, 57eqtr4di 2791 . . . . . . 7 (𝑧 = βŸ¨π‘”, π‘¦βŸ© β†’ ((1st β€˜π‘)β€˜π‘§) = (𝑔(1st β€˜π‘)𝑦))
59 fveq2 6846 . . . . . . . 8 (𝑧 = βŸ¨π‘”, π‘¦βŸ© β†’ ((1st β€˜πΈ)β€˜π‘§) = ((1st β€˜πΈ)β€˜βŸ¨π‘”, π‘¦βŸ©))
60 df-ov 7364 . . . . . . . 8 (𝑔(1st β€˜πΈ)𝑦) = ((1st β€˜πΈ)β€˜βŸ¨π‘”, π‘¦βŸ©)
6159, 60eqtr4di 2791 . . . . . . 7 (𝑧 = βŸ¨π‘”, π‘¦βŸ© β†’ ((1st β€˜πΈ)β€˜π‘§) = (𝑔(1st β€˜πΈ)𝑦))
6258, 61oveq12d 7379 . . . . . 6 (𝑧 = βŸ¨π‘”, π‘¦βŸ© β†’ (((1st β€˜π‘)β€˜π‘§)(Hom β€˜π‘‡)((1st β€˜πΈ)β€˜π‘§)) = ((𝑔(1st β€˜π‘)𝑦)(Hom β€˜π‘‡)(𝑔(1st β€˜πΈ)𝑦)))
6355, 62eleq12d 2828 . . . . 5 (𝑧 = βŸ¨π‘”, π‘¦βŸ© β†’ ((π‘€β€˜π‘§) ∈ (((1st β€˜π‘)β€˜π‘§)(Hom β€˜π‘‡)((1st β€˜πΈ)β€˜π‘§)) ↔ (𝑔𝑀𝑦) ∈ ((𝑔(1st β€˜π‘)𝑦)(Hom β€˜π‘‡)(𝑔(1st β€˜πΈ)𝑦))))
6463ralxp 5801 . . . 4 (βˆ€π‘§ ∈ ((𝑂 Func 𝑆) Γ— 𝐡)(π‘€β€˜π‘§) ∈ (((1st β€˜π‘)β€˜π‘§)(Hom β€˜π‘‡)((1st β€˜πΈ)β€˜π‘§)) ↔ βˆ€π‘” ∈ (𝑂 Func 𝑆)βˆ€π‘¦ ∈ 𝐡 (𝑔𝑀𝑦) ∈ ((𝑔(1st β€˜π‘)𝑦)(Hom β€˜π‘‡)(𝑔(1st β€˜πΈ)𝑦)))
6552, 64sylibr 233 . . 3 (πœ‘ β†’ βˆ€π‘§ ∈ ((𝑂 Func 𝑆) Γ— 𝐡)(π‘€β€˜π‘§) ∈ (((1st β€˜π‘)β€˜π‘§)(Hom β€˜π‘‡)((1st β€˜πΈ)β€˜π‘§)))
66 ovex 7394 . . . . . 6 (𝑂 Func 𝑆) ∈ V
677fvexi 6860 . . . . . 6 𝐡 ∈ V
6866, 67mpoex 8016 . . . . 5 (𝑓 ∈ (𝑂 Func 𝑆), π‘₯ ∈ 𝐡 ↦ (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘₯)(𝑂 Nat 𝑆)𝑓) ↦ ((π‘Žβ€˜π‘₯)β€˜( 1 β€˜π‘₯)))) ∈ V
691, 68eqeltri 2830 . . . 4 𝑀 ∈ V
7069elixp 8848 . . 3 (𝑀 ∈ X𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡)(((1st β€˜π‘)β€˜π‘§)(Hom β€˜π‘‡)((1st β€˜πΈ)β€˜π‘§)) ↔ (𝑀 Fn ((𝑂 Func 𝑆) Γ— 𝐡) ∧ βˆ€π‘§ ∈ ((𝑂 Func 𝑆) Γ— 𝐡)(π‘€β€˜π‘§) ∈ (((1st β€˜π‘)β€˜π‘§)(Hom β€˜π‘‡)((1st β€˜πΈ)β€˜π‘§))))
715, 65, 70sylanbrc 584 . 2 (πœ‘ β†’ 𝑀 ∈ X𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡)(((1st β€˜π‘)β€˜π‘§)(Hom β€˜π‘‡)((1st β€˜πΈ)β€˜π‘§)))
7217adantr 482 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ 𝐢 ∈ Cat)
7319adantr 482 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ 𝑉 ∈ π‘Š)
7421adantr 482 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
7523adantr 482 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (ran (Homf β€˜π‘„) βˆͺ π‘ˆ) βŠ† 𝑉)
76 simpr1 1195 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ 𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡))
77 xp1st 7957 . . . . . 6 (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) β†’ (1st β€˜π‘§) ∈ (𝑂 Func 𝑆))
7876, 77syl 17 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (1st β€˜π‘§) ∈ (𝑂 Func 𝑆))
79 xp2nd 7958 . . . . . 6 (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) β†’ (2nd β€˜π‘§) ∈ 𝐡)
8076, 79syl 17 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (2nd β€˜π‘§) ∈ 𝐡)
81 simpr2 1196 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡))
82 xp1st 7957 . . . . . 6 (𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) β†’ (1st β€˜π‘€) ∈ (𝑂 Func 𝑆))
8381, 82syl 17 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (1st β€˜π‘€) ∈ (𝑂 Func 𝑆))
84 xp2nd 7958 . . . . . 6 (𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) β†’ (2nd β€˜π‘€) ∈ 𝐡)
8581, 84syl 17 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (2nd β€˜π‘€) ∈ 𝐡)
86 simpr3 1197 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))
87 eqid 2733 . . . . . . . . . 10 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
8812, 87fuchom 17857 . . . . . . . . 9 (𝑂 Nat 𝑆) = (Hom β€˜π‘„)
89 eqid 2733 . . . . . . . . 9 (Hom β€˜π‘‚) = (Hom β€˜π‘‚)
90 eqid 2733 . . . . . . . . 9 (Hom β€˜(𝑄 Γ—c 𝑂)) = (Hom β€˜(𝑄 Γ—c 𝑂))
9130, 33, 88, 89, 90, 76, 81xpchom 18076 . . . . . . . 8 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀) = (((1st β€˜π‘§)(𝑂 Nat 𝑆)(1st β€˜π‘€)) Γ— ((2nd β€˜π‘§)(Hom β€˜π‘‚)(2nd β€˜π‘€))))
92 eqid 2733 . . . . . . . . . 10 (Hom β€˜πΆ) = (Hom β€˜πΆ)
9392, 9oppchom 17604 . . . . . . . . 9 ((2nd β€˜π‘§)(Hom β€˜π‘‚)(2nd β€˜π‘€)) = ((2nd β€˜π‘€)(Hom β€˜πΆ)(2nd β€˜π‘§))
9493xpeq2i 5664 . . . . . . . 8 (((1st β€˜π‘§)(𝑂 Nat 𝑆)(1st β€˜π‘€)) Γ— ((2nd β€˜π‘§)(Hom β€˜π‘‚)(2nd β€˜π‘€))) = (((1st β€˜π‘§)(𝑂 Nat 𝑆)(1st β€˜π‘€)) Γ— ((2nd β€˜π‘€)(Hom β€˜πΆ)(2nd β€˜π‘§)))
9591, 94eqtrdi 2789 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀) = (((1st β€˜π‘§)(𝑂 Nat 𝑆)(1st β€˜π‘€)) Γ— ((2nd β€˜π‘€)(Hom β€˜πΆ)(2nd β€˜π‘§))))
9686, 95eleqtrd 2836 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ 𝑔 ∈ (((1st β€˜π‘§)(𝑂 Nat 𝑆)(1st β€˜π‘€)) Γ— ((2nd β€˜π‘€)(Hom β€˜πΆ)(2nd β€˜π‘§))))
97 xp1st 7957 . . . . . 6 (𝑔 ∈ (((1st β€˜π‘§)(𝑂 Nat 𝑆)(1st β€˜π‘€)) Γ— ((2nd β€˜π‘€)(Hom β€˜πΆ)(2nd β€˜π‘§))) β†’ (1st β€˜π‘”) ∈ ((1st β€˜π‘§)(𝑂 Nat 𝑆)(1st β€˜π‘€)))
9896, 97syl 17 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (1st β€˜π‘”) ∈ ((1st β€˜π‘§)(𝑂 Nat 𝑆)(1st β€˜π‘€)))
99 xp2nd 7958 . . . . . 6 (𝑔 ∈ (((1st β€˜π‘§)(𝑂 Nat 𝑆)(1st β€˜π‘€)) Γ— ((2nd β€˜π‘€)(Hom β€˜πΆ)(2nd β€˜π‘§))) β†’ (2nd β€˜π‘”) ∈ ((2nd β€˜π‘€)(Hom β€˜πΆ)(2nd β€˜π‘§)))
10096, 99syl 17 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (2nd β€˜π‘”) ∈ ((2nd β€˜π‘€)(Hom β€˜πΆ)(2nd β€˜π‘§)))
1016, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 72, 73, 74, 75, 78, 80, 83, 85, 98, 100, 1yonedalem3b 18176 . . . 4 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (((1st β€˜π‘€)𝑀(2nd β€˜π‘€))(⟨((1st β€˜π‘§)(1st β€˜π‘)(2nd β€˜π‘§)), ((1st β€˜π‘€)(1st β€˜π‘)(2nd β€˜π‘€))⟩(compβ€˜π‘‡)((1st β€˜π‘€)(1st β€˜πΈ)(2nd β€˜π‘€)))((1st β€˜π‘”)(⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜π‘)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)(2nd β€˜π‘”))) = (((1st β€˜π‘”)(⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)(2nd β€˜π‘”))(⟨((1st β€˜π‘§)(1st β€˜π‘)(2nd β€˜π‘§)), ((1st β€˜π‘§)(1st β€˜πΈ)(2nd β€˜π‘§))⟩(compβ€˜π‘‡)((1st β€˜π‘€)(1st β€˜πΈ)(2nd β€˜π‘€)))((1st β€˜π‘§)𝑀(2nd β€˜π‘§))))
102 1st2nd2 7964 . . . . . . . . . 10 (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) β†’ 𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
10376, 102syl 17 . . . . . . . . 9 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ 𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
104103fveq2d 6850 . . . . . . . 8 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((1st β€˜π‘)β€˜π‘§) = ((1st β€˜π‘)β€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩))
105 df-ov 7364 . . . . . . . 8 ((1st β€˜π‘§)(1st β€˜π‘)(2nd β€˜π‘§)) = ((1st β€˜π‘)β€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
106104, 105eqtr4di 2791 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((1st β€˜π‘)β€˜π‘§) = ((1st β€˜π‘§)(1st β€˜π‘)(2nd β€˜π‘§)))
107 1st2nd2 7964 . . . . . . . . . 10 (𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) β†’ 𝑀 = ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)
10881, 107syl 17 . . . . . . . . 9 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ 𝑀 = ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)
109108fveq2d 6850 . . . . . . . 8 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((1st β€˜π‘)β€˜π‘€) = ((1st β€˜π‘)β€˜βŸ¨(1st β€˜π‘€), (2nd β€˜π‘€)⟩))
110 df-ov 7364 . . . . . . . 8 ((1st β€˜π‘€)(1st β€˜π‘)(2nd β€˜π‘€)) = ((1st β€˜π‘)β€˜βŸ¨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)
111109, 110eqtr4di 2791 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((1st β€˜π‘)β€˜π‘€) = ((1st β€˜π‘€)(1st β€˜π‘)(2nd β€˜π‘€)))
112106, 111opeq12d 4842 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜π‘)β€˜π‘€)⟩ = ⟨((1st β€˜π‘§)(1st β€˜π‘)(2nd β€˜π‘§)), ((1st β€˜π‘€)(1st β€˜π‘)(2nd β€˜π‘€))⟩)
113108fveq2d 6850 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((1st β€˜πΈ)β€˜π‘€) = ((1st β€˜πΈ)β€˜βŸ¨(1st β€˜π‘€), (2nd β€˜π‘€)⟩))
114 df-ov 7364 . . . . . . 7 ((1st β€˜π‘€)(1st β€˜πΈ)(2nd β€˜π‘€)) = ((1st β€˜πΈ)β€˜βŸ¨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)
115113, 114eqtr4di 2791 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((1st β€˜πΈ)β€˜π‘€) = ((1st β€˜π‘€)(1st β€˜πΈ)(2nd β€˜π‘€)))
116112, 115oveq12d 7379 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜π‘)β€˜π‘€)⟩(compβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘€)) = (⟨((1st β€˜π‘§)(1st β€˜π‘)(2nd β€˜π‘§)), ((1st β€˜π‘€)(1st β€˜π‘)(2nd β€˜π‘€))⟩(compβ€˜π‘‡)((1st β€˜π‘€)(1st β€˜πΈ)(2nd β€˜π‘€))))
117108fveq2d 6850 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (π‘€β€˜π‘€) = (π‘€β€˜βŸ¨(1st β€˜π‘€), (2nd β€˜π‘€)⟩))
118 df-ov 7364 . . . . . 6 ((1st β€˜π‘€)𝑀(2nd β€˜π‘€)) = (π‘€β€˜βŸ¨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)
119117, 118eqtr4di 2791 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (π‘€β€˜π‘€) = ((1st β€˜π‘€)𝑀(2nd β€˜π‘€)))
120103, 108oveq12d 7379 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (𝑧(2nd β€˜π‘)𝑀) = (⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜π‘)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩))
121 1st2nd2 7964 . . . . . . . 8 (𝑔 ∈ (((1st β€˜π‘§)(𝑂 Nat 𝑆)(1st β€˜π‘€)) Γ— ((2nd β€˜π‘€)(Hom β€˜πΆ)(2nd β€˜π‘§))) β†’ 𝑔 = ⟨(1st β€˜π‘”), (2nd β€˜π‘”)⟩)
12296, 121syl 17 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ 𝑔 = ⟨(1st β€˜π‘”), (2nd β€˜π‘”)⟩)
123120, 122fveq12d 6853 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((𝑧(2nd β€˜π‘)𝑀)β€˜π‘”) = ((⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜π‘)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)β€˜βŸ¨(1st β€˜π‘”), (2nd β€˜π‘”)⟩))
124 df-ov 7364 . . . . . 6 ((1st β€˜π‘”)(⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜π‘)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)(2nd β€˜π‘”)) = ((⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜π‘)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)β€˜βŸ¨(1st β€˜π‘”), (2nd β€˜π‘”)⟩)
125123, 124eqtr4di 2791 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((𝑧(2nd β€˜π‘)𝑀)β€˜π‘”) = ((1st β€˜π‘”)(⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜π‘)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)(2nd β€˜π‘”)))
126116, 119, 125oveq123d 7382 . . . 4 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((π‘€β€˜π‘€)(⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜π‘)β€˜π‘€)⟩(compβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘€))((𝑧(2nd β€˜π‘)𝑀)β€˜π‘”)) = (((1st β€˜π‘€)𝑀(2nd β€˜π‘€))(⟨((1st β€˜π‘§)(1st β€˜π‘)(2nd β€˜π‘§)), ((1st β€˜π‘€)(1st β€˜π‘)(2nd β€˜π‘€))⟩(compβ€˜π‘‡)((1st β€˜π‘€)(1st β€˜πΈ)(2nd β€˜π‘€)))((1st β€˜π‘”)(⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜π‘)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)(2nd β€˜π‘”))))
127103fveq2d 6850 . . . . . . . 8 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((1st β€˜πΈ)β€˜π‘§) = ((1st β€˜πΈ)β€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩))
128 df-ov 7364 . . . . . . . 8 ((1st β€˜π‘§)(1st β€˜πΈ)(2nd β€˜π‘§)) = ((1st β€˜πΈ)β€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
129127, 128eqtr4di 2791 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((1st β€˜πΈ)β€˜π‘§) = ((1st β€˜π‘§)(1st β€˜πΈ)(2nd β€˜π‘§)))
130106, 129opeq12d 4842 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜πΈ)β€˜π‘§)⟩ = ⟨((1st β€˜π‘§)(1st β€˜π‘)(2nd β€˜π‘§)), ((1st β€˜π‘§)(1st β€˜πΈ)(2nd β€˜π‘§))⟩)
131130, 115oveq12d 7379 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜πΈ)β€˜π‘§)⟩(compβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘€)) = (⟨((1st β€˜π‘§)(1st β€˜π‘)(2nd β€˜π‘§)), ((1st β€˜π‘§)(1st β€˜πΈ)(2nd β€˜π‘§))⟩(compβ€˜π‘‡)((1st β€˜π‘€)(1st β€˜πΈ)(2nd β€˜π‘€))))
132103, 108oveq12d 7379 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (𝑧(2nd β€˜πΈ)𝑀) = (⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩))
133132, 122fveq12d 6853 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((𝑧(2nd β€˜πΈ)𝑀)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‘”)⟩))
134 df-ov 7364 . . . . . 6 ((1st β€˜π‘”)(⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘€), (2nd 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‘”)⟩)
135133, 134eqtr4di 2791 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((𝑧(2nd β€˜πΈ)𝑀)β€˜π‘”) = ((1st β€˜π‘”)(⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)(2nd β€˜π‘”)))
136103fveq2d 6850 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (π‘€β€˜π‘§) = (π‘€β€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩))
137 df-ov 7364 . . . . . 6 ((1st β€˜π‘§)𝑀(2nd β€˜π‘§)) = (π‘€β€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
138136, 137eqtr4di 2791 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (π‘€β€˜π‘§) = ((1st β€˜π‘§)𝑀(2nd β€˜π‘§)))
139131, 135, 138oveq123d 7382 . . . 4 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (((𝑧(2nd β€˜πΈ)𝑀)β€˜π‘”)(⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜πΈ)β€˜π‘§)⟩(compβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘€))(π‘€β€˜π‘§)) = (((1st β€˜π‘”)(⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)(2nd β€˜π‘”))(⟨((1st β€˜π‘§)(1st β€˜π‘)(2nd β€˜π‘§)), ((1st β€˜π‘§)(1st β€˜πΈ)(2nd β€˜π‘§))⟩(compβ€˜π‘‡)((1st β€˜π‘€)(1st β€˜πΈ)(2nd β€˜π‘€)))((1st β€˜π‘§)𝑀(2nd β€˜π‘§))))
140101, 126, 1393eqtr4d 2783 . . 3 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((π‘€β€˜π‘€)(⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜π‘)β€˜π‘€)⟩(compβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘€))((𝑧(2nd β€˜π‘)𝑀)β€˜π‘”)) = (((𝑧(2nd β€˜πΈ)𝑀)β€˜π‘”)(⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜πΈ)β€˜π‘§)⟩(compβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘€))(π‘€β€˜π‘§)))
141140ralrimivvva 3197 . 2 (πœ‘ β†’ βˆ€π‘§ ∈ ((𝑂 Func 𝑆) Γ— 𝐡)βˆ€π‘€ ∈ ((𝑂 Func 𝑆) Γ— 𝐡)βˆ€π‘” ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀)((π‘€β€˜π‘€)(⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜π‘)β€˜π‘€)⟩(compβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘€))((𝑧(2nd β€˜π‘)𝑀)β€˜π‘”)) = (((𝑧(2nd β€˜πΈ)𝑀)β€˜π‘”)(⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜πΈ)β€˜π‘§)⟩(compβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘€))(π‘€β€˜π‘§)))
142 eqid 2733 . . 3 ((𝑄 Γ—c 𝑂) Nat 𝑇) = ((𝑄 Γ—c 𝑂) Nat 𝑇)
143 eqid 2733 . . 3 (compβ€˜π‘‡) = (compβ€˜π‘‡)
144142, 33, 90, 29, 143, 37, 44isnat2 17843 . 2 (πœ‘ β†’ (𝑀 ∈ (𝑍((𝑄 Γ—c 𝑂) Nat 𝑇)𝐸) ↔ (𝑀 ∈ X𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡)(((1st β€˜π‘)β€˜π‘§)(Hom β€˜π‘‡)((1st β€˜πΈ)β€˜π‘§)) ∧ βˆ€π‘§ ∈ ((𝑂 Func 𝑆) Γ— 𝐡)βˆ€π‘€ ∈ ((𝑂 Func 𝑆) Γ— 𝐡)βˆ€π‘” ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀)((π‘€β€˜π‘€)(⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜π‘)β€˜π‘€)⟩(compβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘€))((𝑧(2nd β€˜π‘)𝑀)β€˜π‘”)) = (((𝑧(2nd β€˜πΈ)𝑀)β€˜π‘”)(⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜πΈ)β€˜π‘§)⟩(compβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘€))(π‘€β€˜π‘§)))))
14571, 141, 144mpbir2and 712 1 (πœ‘ β†’ 𝑀 ∈ (𝑍((𝑄 Γ—c 𝑂) Nat 𝑇)𝐸))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  Vcvv 3447   βˆͺ cun 3912   βŠ† wss 3914  βŸ¨cop 4596   class class class wbr 5109   ↦ cmpt 5192   Γ— cxp 5635  ran crn 5638  Rel wrel 5642   Fn wfn 6495  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363  1st c1st 7923  2nd c2nd 7924  tpos ctpos 8160  Xcixp 8841  Basecbs 17091  Hom chom 17152  compcco 17153  Catccat 17552  Idccid 17553  Homf chomf 17554  oppCatcoppc 17599   Func cfunc 17748   ∘func ccofu 17750   Nat cnat 17836   FuncCat cfuc 17837  SetCatcsetc 17969   Γ—c cxpc 18064   1stF c1stf 18065   2ndF c2ndf 18066   ⟨,⟩F cprf 18067   evalF cevlf 18106  HomFchof 18145  Yoncyon 18146
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
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 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-tpos 8161  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-pm 8774  df-ixp 8842  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-8 12230  df-9 12231  df-n0 12422  df-z 12508  df-dec 12627  df-uz 12772  df-fz 13434  df-struct 17027  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-hom 17165  df-cco 17166  df-cat 17556  df-cid 17557  df-homf 17558  df-comf 17559  df-oppc 17600  df-ssc 17701  df-resc 17702  df-subc 17703  df-func 17752  df-cofu 17754  df-nat 17838  df-fuc 17839  df-setc 17970  df-xpc 18068  df-1stf 18069  df-2ndf 18070  df-prf 18071  df-evlf 18110  df-curf 18111  df-hof 18147  df-yon 18148
This theorem is referenced by:  yonedainv  18178
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