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

Theorem invfuc 16681
Description: If 𝑉(𝑥) is an inverse to 𝑈(𝑥) for each 𝑥, and 𝑈 is a natural transformation, then 𝑉 is also a natural transformation, and they are inverse in the functor category. (Contributed by Mario Carneiro, 28-Jan-2017.)
Hypotheses
Ref Expression
fuciso.q 𝑄 = (𝐶 FuncCat 𝐷)
fuciso.b 𝐵 = (Base‘𝐶)
fuciso.n 𝑁 = (𝐶 Nat 𝐷)
fuciso.f (𝜑𝐹 ∈ (𝐶 Func 𝐷))
fuciso.g (𝜑𝐺 ∈ (𝐶 Func 𝐷))
fucinv.i 𝐼 = (Inv‘𝑄)
fucinv.j 𝐽 = (Inv‘𝐷)
invfuc.u (𝜑𝑈 ∈ (𝐹𝑁𝐺))
invfuc.v ((𝜑𝑥𝐵) → (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))𝑋)
Assertion
Ref Expression
invfuc (𝜑𝑈(𝐹𝐼𝐺)(𝑥𝐵𝑋))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷   𝑥,𝐼   𝑥,𝐹   𝑥,𝐺   𝑥,𝐽   𝑥,𝑁   𝜑,𝑥   𝑥,𝑄   𝑥,𝑈
Allowed substitution hint:   𝑋(𝑥)

Proof of Theorem invfuc
Dummy variables 𝑦 𝑓 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 invfuc.u . 2 (𝜑𝑈 ∈ (𝐹𝑁𝐺))
2 invfuc.v . . . . . . . 8 ((𝜑𝑥𝐵) → (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))𝑋)
3 eqid 2651 . . . . . . . . . 10 (Base‘𝐷) = (Base‘𝐷)
4 fucinv.j . . . . . . . . . 10 𝐽 = (Inv‘𝐷)
5 fuciso.f . . . . . . . . . . . . 13 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
6 funcrcl 16570 . . . . . . . . . . . . 13 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
75, 6syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
87simprd 478 . . . . . . . . . . 11 (𝜑𝐷 ∈ Cat)
98adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝐵) → 𝐷 ∈ Cat)
10 fuciso.b . . . . . . . . . . . 12 𝐵 = (Base‘𝐶)
11 relfunc 16569 . . . . . . . . . . . . 13 Rel (𝐶 Func 𝐷)
12 1st2ndbr 7261 . . . . . . . . . . . . 13 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1311, 5, 12sylancr 696 . . . . . . . . . . . 12 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1410, 3, 13funcf1 16573 . . . . . . . . . . 11 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
1514ffvelrnda 6399 . . . . . . . . . 10 ((𝜑𝑥𝐵) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
16 fuciso.g . . . . . . . . . . . . 13 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
17 1st2ndbr 7261 . . . . . . . . . . . . 13 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
1811, 16, 17sylancr 696 . . . . . . . . . . . 12 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
1910, 3, 18funcf1 16573 . . . . . . . . . . 11 (𝜑 → (1st𝐺):𝐵⟶(Base‘𝐷))
2019ffvelrnda 6399 . . . . . . . . . 10 ((𝜑𝑥𝐵) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
21 eqid 2651 . . . . . . . . . 10 (Hom ‘𝐷) = (Hom ‘𝐷)
223, 4, 9, 15, 20, 21invss 16468 . . . . . . . . 9 ((𝜑𝑥𝐵) → (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) ⊆ ((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)) × (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥))))
2322ssbrd 4728 . . . . . . . 8 ((𝜑𝑥𝐵) → ((𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))𝑋 → (𝑈𝑥)((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)) × (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))𝑋))
242, 23mpd 15 . . . . . . 7 ((𝜑𝑥𝐵) → (𝑈𝑥)((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)) × (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))𝑋)
25 brxp 5181 . . . . . . . 8 ((𝑈𝑥)((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)) × (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))𝑋 ↔ ((𝑈𝑥) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)) ∧ 𝑋 ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥))))
2625simprbi 479 . . . . . . 7 ((𝑈𝑥)((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)) × (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))𝑋𝑋 ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
2724, 26syl 17 . . . . . 6 ((𝜑𝑥𝐵) → 𝑋 ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
2827ralrimiva 2995 . . . . 5 (𝜑 → ∀𝑥𝐵 𝑋 ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
29 fvex 6239 . . . . . . 7 (Base‘𝐶) ∈ V
3010, 29eqeltri 2726 . . . . . 6 𝐵 ∈ V
31 mptelixpg 7987 . . . . . 6 (𝐵 ∈ V → ((𝑥𝐵𝑋) ∈ X𝑥𝐵 (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)) ↔ ∀𝑥𝐵 𝑋 ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥))))
3230, 31ax-mp 5 . . . . 5 ((𝑥𝐵𝑋) ∈ X𝑥𝐵 (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)) ↔ ∀𝑥𝐵 𝑋 ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
3328, 32sylibr 224 . . . 4 (𝜑 → (𝑥𝐵𝑋) ∈ X𝑥𝐵 (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
34 fveq2 6229 . . . . . 6 (𝑥 = 𝑦 → ((1st𝐺)‘𝑥) = ((1st𝐺)‘𝑦))
35 fveq2 6229 . . . . . 6 (𝑥 = 𝑦 → ((1st𝐹)‘𝑥) = ((1st𝐹)‘𝑦))
3634, 35oveq12d 6708 . . . . 5 (𝑥 = 𝑦 → (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)) = (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
3736cbvixpv 7968 . . . 4 X𝑥𝐵 (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)) = X𝑦𝐵 (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦))
3833, 37syl6eleq 2740 . . 3 (𝜑 → (𝑥𝐵𝑋) ∈ X𝑦𝐵 (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
39 simpr2 1088 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧𝐵)
40 simpr 476 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐵) → 𝑥𝐵)
41 eqid 2651 . . . . . . . . . . . . . . . . . . 19 (𝑥𝐵𝑋) = (𝑥𝐵𝑋)
4241fvmpt2 6330 . . . . . . . . . . . . . . . . . 18 ((𝑥𝐵𝑋 ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥))) → ((𝑥𝐵𝑋)‘𝑥) = 𝑋)
4340, 27, 42syl2anc 694 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐵) → ((𝑥𝐵𝑋)‘𝑥) = 𝑋)
442, 43breqtrrd 4713 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐵) → (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥))
4544ralrimiva 2995 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥))
4645adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥))
47 nfcv 2793 . . . . . . . . . . . . . . . 16 𝑥(𝑈𝑧)
48 nfcv 2793 . . . . . . . . . . . . . . . 16 𝑥(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))
49 nffvmpt1 6237 . . . . . . . . . . . . . . . 16 𝑥((𝑥𝐵𝑋)‘𝑧)
5047, 48, 49nfbr 4732 . . . . . . . . . . . . . . 15 𝑥(𝑈𝑧)(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧)
51 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (𝑈𝑥) = (𝑈𝑧))
52 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ((1st𝐹)‘𝑥) = ((1st𝐹)‘𝑧))
53 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ((1st𝐺)‘𝑥) = ((1st𝐺)‘𝑧))
5452, 53oveq12d 6708 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) = (((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧)))
55 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → ((𝑥𝐵𝑋)‘𝑥) = ((𝑥𝐵𝑋)‘𝑧))
5651, 54, 55breq123d 4699 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → ((𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) ↔ (𝑈𝑧)(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧)))
5750, 56rspc 3334 . . . . . . . . . . . . . 14 (𝑧𝐵 → (∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) → (𝑈𝑧)(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧)))
5839, 46, 57sylc 65 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑧)(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧))
598adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐷 ∈ Cat)
6014adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐹):𝐵⟶(Base‘𝐷))
6160, 39ffvelrnd 6400 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐹)‘𝑧) ∈ (Base‘𝐷))
6219adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐺):𝐵⟶(Base‘𝐷))
6362, 39ffvelrnd 6400 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐺)‘𝑧) ∈ (Base‘𝐷))
64 eqid 2651 . . . . . . . . . . . . . 14 (Sect‘𝐷) = (Sect‘𝐷)
653, 4, 59, 61, 63, 64isinv 16467 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈𝑧)(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧) ↔ ((𝑈𝑧)(((1st𝐹)‘𝑧)(Sect‘𝐷)((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧) ∧ ((𝑥𝐵𝑋)‘𝑧)(((1st𝐺)‘𝑧)(Sect‘𝐷)((1st𝐹)‘𝑧))(𝑈𝑧))))
6658, 65mpbid 222 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈𝑧)(((1st𝐹)‘𝑧)(Sect‘𝐷)((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧) ∧ ((𝑥𝐵𝑋)‘𝑧)(((1st𝐺)‘𝑧)(Sect‘𝐷)((1st𝐹)‘𝑧))(𝑈𝑧)))
6766simpld 474 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑧)(((1st𝐹)‘𝑧)(Sect‘𝐷)((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧))
68 eqid 2651 . . . . . . . . . . . 12 (comp‘𝐷) = (comp‘𝐷)
69 eqid 2651 . . . . . . . . . . . 12 (Id‘𝐷) = (Id‘𝐷)
703, 21, 68, 69, 64, 59, 61, 63issect 16460 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈𝑧)(((1st𝐹)‘𝑧)(Sect‘𝐷)((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧) ↔ ((𝑈𝑧) ∈ (((1st𝐹)‘𝑧)(Hom ‘𝐷)((1st𝐺)‘𝑧)) ∧ ((𝑥𝐵𝑋)‘𝑧) ∈ (((1st𝐺)‘𝑧)(Hom ‘𝐷)((1st𝐹)‘𝑧)) ∧ (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(𝑈𝑧)) = ((Id‘𝐷)‘((1st𝐹)‘𝑧)))))
7167, 70mpbid 222 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈𝑧) ∈ (((1st𝐹)‘𝑧)(Hom ‘𝐷)((1st𝐺)‘𝑧)) ∧ ((𝑥𝐵𝑋)‘𝑧) ∈ (((1st𝐺)‘𝑧)(Hom ‘𝐷)((1st𝐹)‘𝑧)) ∧ (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(𝑈𝑧)) = ((Id‘𝐷)‘((1st𝐹)‘𝑧))))
7271simp3d 1095 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(𝑈𝑧)) = ((Id‘𝐷)‘((1st𝐹)‘𝑧)))
7372oveq1d 6705 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(𝑈𝑧))(⟨((1st𝐹)‘𝑦), ((1st𝐹)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑦(2nd𝐹)𝑧)‘𝑓)) = (((Id‘𝐷)‘((1st𝐹)‘𝑧))(⟨((1st𝐹)‘𝑦), ((1st𝐹)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑦(2nd𝐹)𝑧)‘𝑓)))
74 simpr1 1087 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦𝐵)
7560, 74ffvelrnd 6400 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
76 eqid 2651 . . . . . . . . . . 11 (Hom ‘𝐶) = (Hom ‘𝐶)
7713adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
7810, 76, 21, 77, 74, 39funcf2 16575 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑦(2nd𝐹)𝑧):(𝑦(Hom ‘𝐶)𝑧)⟶(((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑧)))
79 simpr3 1089 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))
8078, 79ffvelrnd 6400 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2nd𝐹)𝑧)‘𝑓) ∈ (((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑧)))
813, 21, 69, 59, 75, 68, 61, 80catlid 16391 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((Id‘𝐷)‘((1st𝐹)‘𝑧))(⟨((1st𝐹)‘𝑦), ((1st𝐹)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑦(2nd𝐹)𝑧)‘𝑓)) = ((𝑦(2nd𝐹)𝑧)‘𝑓))
8273, 81eqtr2d 2686 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2nd𝐹)𝑧)‘𝑓) = ((((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(𝑈𝑧))(⟨((1st𝐹)‘𝑦), ((1st𝐹)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑦(2nd𝐹)𝑧)‘𝑓)))
83 fuciso.n . . . . . . . . 9 𝑁 = (𝐶 Nat 𝐷)
841adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑈 ∈ (𝐹𝑁𝐺))
8583, 84nat1st2nd 16658 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑈 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
8683, 85, 10, 21, 39natcl 16660 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑧) ∈ (((1st𝐹)‘𝑧)(Hom ‘𝐷)((1st𝐺)‘𝑧)))
8771simp2d 1094 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥𝐵𝑋)‘𝑧) ∈ (((1st𝐺)‘𝑧)(Hom ‘𝐷)((1st𝐹)‘𝑧)))
883, 21, 68, 59, 75, 61, 63, 80, 86, 61, 87catass 16394 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(𝑈𝑧))(⟨((1st𝐹)‘𝑦), ((1st𝐹)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑦(2nd𝐹)𝑧)‘𝑓)) = (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑈𝑧)(⟨((1st𝐹)‘𝑦), ((1st𝐹)‘𝑧)⟩(comp‘𝐷)((1st𝐺)‘𝑧))((𝑦(2nd𝐹)𝑧)‘𝑓))))
8983, 85, 10, 76, 68, 74, 39, 79nati 16662 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈𝑧)(⟨((1st𝐹)‘𝑦), ((1st𝐹)‘𝑧)⟩(comp‘𝐷)((1st𝐺)‘𝑧))((𝑦(2nd𝐹)𝑧)‘𝑓)) = (((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))(𝑈𝑦)))
9089oveq2d 6706 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑈𝑧)(⟨((1st𝐹)‘𝑦), ((1st𝐹)‘𝑧)⟩(comp‘𝐷)((1st𝐺)‘𝑧))((𝑦(2nd𝐹)𝑧)‘𝑓))) = (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))(𝑈𝑦))))
9182, 88, 903eqtrd 2689 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2nd𝐹)𝑧)‘𝑓) = (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))(𝑈𝑦))))
9291oveq1d 6705 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑦(2nd𝐹)𝑧)‘𝑓)(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥𝐵𝑋)‘𝑦)) = ((((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))(𝑈𝑦)))(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥𝐵𝑋)‘𝑦)))
9362, 74ffvelrnd 6400 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐺)‘𝑦) ∈ (Base‘𝐷))
94 nfcv 2793 . . . . . . . . . . . . 13 𝑥(𝑈𝑦)
95 nfcv 2793 . . . . . . . . . . . . 13 𝑥(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))
96 nffvmpt1 6237 . . . . . . . . . . . . 13 𝑥((𝑥𝐵𝑋)‘𝑦)
9794, 95, 96nfbr 4732 . . . . . . . . . . . 12 𝑥(𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)
98 fveq2 6229 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑈𝑥) = (𝑈𝑦))
9935, 34oveq12d 6708 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) = (((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦)))
100 fveq2 6229 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑥𝐵𝑋)‘𝑥) = ((𝑥𝐵𝑋)‘𝑦))
10198, 99, 100breq123d 4699 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) ↔ (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)))
10297, 101rspc 3334 . . . . . . . . . . 11 (𝑦𝐵 → (∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) → (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)))
10374, 46, 102sylc 65 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦))
1043, 4, 59, 75, 93, 64isinv 16467 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦) ↔ ((𝑈𝑦)(((1st𝐹)‘𝑦)(Sect‘𝐷)((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦) ∧ ((𝑥𝐵𝑋)‘𝑦)(((1st𝐺)‘𝑦)(Sect‘𝐷)((1st𝐹)‘𝑦))(𝑈𝑦))))
105103, 104mpbid 222 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈𝑦)(((1st𝐹)‘𝑦)(Sect‘𝐷)((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦) ∧ ((𝑥𝐵𝑋)‘𝑦)(((1st𝐺)‘𝑦)(Sect‘𝐷)((1st𝐹)‘𝑦))(𝑈𝑦)))
106105simprd 478 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥𝐵𝑋)‘𝑦)(((1st𝐺)‘𝑦)(Sect‘𝐷)((1st𝐹)‘𝑦))(𝑈𝑦))
1073, 21, 68, 69, 64, 59, 93, 75issect 16460 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑥𝐵𝑋)‘𝑦)(((1st𝐺)‘𝑦)(Sect‘𝐷)((1st𝐹)‘𝑦))(𝑈𝑦) ↔ (((𝑥𝐵𝑋)‘𝑦) ∈ (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦)) ∧ (𝑈𝑦) ∈ (((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑦)) ∧ ((𝑈𝑦)(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)) = ((Id‘𝐷)‘((1st𝐺)‘𝑦)))))
108106, 107mpbid 222 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑥𝐵𝑋)‘𝑦) ∈ (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦)) ∧ (𝑈𝑦) ∈ (((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑦)) ∧ ((𝑈𝑦)(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)) = ((Id‘𝐷)‘((1st𝐺)‘𝑦))))
109108simp1d 1093 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥𝐵𝑋)‘𝑦) ∈ (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
110108simp2d 1094 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑦) ∈ (((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑦)))
11118adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
11210, 76, 21, 111, 74, 39funcf2 16575 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑦(2nd𝐺)𝑧):(𝑦(Hom ‘𝐶)𝑧)⟶(((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑧)))
113112, 79ffvelrnd 6400 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2nd𝐺)𝑧)‘𝑓) ∈ (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑧)))
1143, 21, 68, 59, 75, 93, 63, 110, 113catcocl 16393 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))(𝑈𝑦)) ∈ (((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑧)))
1153, 21, 68, 59, 93, 75, 63, 109, 114, 61, 87catass 16394 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))(𝑈𝑦)))(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥𝐵𝑋)‘𝑦)) = (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐺)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))(𝑈𝑦))(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑦))))
11683, 85, 10, 21, 74natcl 16660 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑦) ∈ (((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑦)))
1173, 21, 68, 59, 93, 75, 93, 109, 116, 63, 113catass 16394 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))(𝑈𝑦))(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑦)) = (((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐺)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))((𝑈𝑦)(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦))))
118108simp3d 1095 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈𝑦)(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)) = ((Id‘𝐷)‘((1st𝐺)‘𝑦)))
119118oveq2d 6706 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐺)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))((𝑈𝑦)(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦))) = (((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐺)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))((Id‘𝐷)‘((1st𝐺)‘𝑦))))
1203, 21, 69, 59, 93, 68, 63, 113catrid 16392 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐺)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))((Id‘𝐷)‘((1st𝐺)‘𝑦))) = ((𝑦(2nd𝐺)𝑧)‘𝑓))
121117, 119, 1203eqtrd 2689 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))(𝑈𝑦))(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑦)) = ((𝑦(2nd𝐺)𝑧)‘𝑓))
122121oveq2d 6706 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐺)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))(𝑈𝑦))(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑦))) = (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐺)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑦(2nd𝐺)𝑧)‘𝑓)))
12392, 115, 1223eqtrrd 2690 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐺)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑦(2nd𝐺)𝑧)‘𝑓)) = (((𝑦(2nd𝐹)𝑧)‘𝑓)(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥𝐵𝑋)‘𝑦)))
124123ralrimivvva 3001 . . 3 (𝜑 → ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧)(((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐺)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑦(2nd𝐺)𝑧)‘𝑓)) = (((𝑦(2nd𝐹)𝑧)‘𝑓)(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥𝐵𝑋)‘𝑦)))
12583, 10, 76, 21, 68, 16, 5isnat2 16655 . . 3 (𝜑 → ((𝑥𝐵𝑋) ∈ (𝐺𝑁𝐹) ↔ ((𝑥𝐵𝑋) ∈ X𝑦𝐵 (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦)) ∧ ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧)(((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐺)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑦(2nd𝐺)𝑧)‘𝑓)) = (((𝑦(2nd𝐹)𝑧)‘𝑓)(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥𝐵𝑋)‘𝑦)))))
12638, 124, 125mpbir2and 977 . 2 (𝜑 → (𝑥𝐵𝑋) ∈ (𝐺𝑁𝐹))
127 nfv 1883 . . . 4 𝑦(𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥)
128127, 97, 101cbvral 3197 . . 3 (∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) ↔ ∀𝑦𝐵 (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦))
12945, 128sylib 208 . 2 (𝜑 → ∀𝑦𝐵 (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦))
130 fuciso.q . . 3 𝑄 = (𝐶 FuncCat 𝐷)
131 fucinv.i . . 3 𝐼 = (Inv‘𝑄)
132130, 10, 83, 5, 16, 131, 4fucinv 16680 . 2 (𝜑 → (𝑈(𝐹𝐼𝐺)(𝑥𝐵𝑋) ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ (𝑥𝐵𝑋) ∈ (𝐺𝑁𝐹) ∧ ∀𝑦𝐵 (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦))))
1331, 126, 129, 132mpbir3and 1264 1 (𝜑𝑈(𝐹𝐼𝐺)(𝑥𝐵𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cop 4216   class class class wbr 4685  cmpt 4762   × cxp 5141  Rel wrel 5148  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  Xcixp 7950  Basecbs 15904  Hom chom 15999  compcco 16000  Catccat 16372  Idccid 16373  Sectcsect 16451  Invcinv 16452   Func cfunc 16561   Nat cnat 16648   FuncCat cfuc 16649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-hom 16013  df-cco 16014  df-cat 16376  df-cid 16377  df-sect 16454  df-inv 16455  df-func 16565  df-nat 16650  df-fuc 16651
This theorem is referenced by:  fuciso  16682  yonedainv  16968
  Copyright terms: Public domain W3C validator