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

Theorem invfuc 16457
 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 2610 . . . . . . . . . 10 (Base‘𝐷) = (Base‘𝐷)
4 fucinv.j . . . . . . . . . 10 𝐽 = (Inv‘𝐷)
5 fuciso.f . . . . . . . . . . . . 13 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
6 funcrcl 16346 . . . . . . . . . . . . 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 16345 . . . . . . . . . . . . 13 Rel (𝐶 Func 𝐷)
12 1st2ndbr 7108 . . . . . . . . . . . . 13 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1311, 5, 12sylancr 694 . . . . . . . . . . . 12 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1410, 3, 13funcf1 16349 . . . . . . . . . . 11 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
1514ffvelrnda 6267 . . . . . . . . . 10 ((𝜑𝑥𝐵) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
16 fuciso.g . . . . . . . . . . . . 13 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
17 1st2ndbr 7108 . . . . . . . . . . . . 13 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
1811, 16, 17sylancr 694 . . . . . . . . . . . 12 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
1910, 3, 18funcf1 16349 . . . . . . . . . . 11 (𝜑 → (1st𝐺):𝐵⟶(Base‘𝐷))
2019ffvelrnda 6267 . . . . . . . . . 10 ((𝜑𝑥𝐵) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
21 eqid 2610 . . . . . . . . . 10 (Hom ‘𝐷) = (Hom ‘𝐷)
223, 4, 9, 15, 20, 21invss 16244 . . . . . . . . 9 ((𝜑𝑥𝐵) → (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) ⊆ ((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)) × (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥))))
2322ssbrd 4626 . . . . . . . 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 5071 . . . . . . . 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 2949 . . . . 5 (𝜑 → ∀𝑥𝐵 𝑋 ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
29 fvex 6113 . . . . . . 7 (Base‘𝐶) ∈ V
3010, 29eqeltri 2684 . . . . . 6 𝐵 ∈ V
31 mptelixpg 7831 . . . . . 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 223 . . . 4 (𝜑 → (𝑥𝐵𝑋) ∈ X𝑥𝐵 (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
34 fveq2 6103 . . . . . 6 (𝑥 = 𝑦 → ((1st𝐺)‘𝑥) = ((1st𝐺)‘𝑦))
35 fveq2 6103 . . . . . 6 (𝑥 = 𝑦 → ((1st𝐹)‘𝑥) = ((1st𝐹)‘𝑦))
3634, 35oveq12d 6567 . . . . 5 (𝑥 = 𝑦 → (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)) = (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
3736cbvixpv 7812 . . . 4 X𝑥𝐵 (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)) = X𝑦𝐵 (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦))
3833, 37syl6eleq 2698 . . 3 (𝜑 → (𝑥𝐵𝑋) ∈ X𝑦𝐵 (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
39 simpr2 1061 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧𝐵)
40 simpr 476 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐵) → 𝑥𝐵)
41 eqid 2610 . . . . . . . . . . . . . . . . . . 19 (𝑥𝐵𝑋) = (𝑥𝐵𝑋)
4241fvmpt2 6200 . . . . . . . . . . . . . . . . . 18 ((𝑥𝐵𝑋 ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥))) → ((𝑥𝐵𝑋)‘𝑥) = 𝑋)
4340, 27, 42syl2anc 691 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐵) → ((𝑥𝐵𝑋)‘𝑥) = 𝑋)
442, 43breqtrrd 4611 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐵) → (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥))
4544ralrimiva 2949 . . . . . . . . . . . . . . 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 2751 . . . . . . . . . . . . . . . 16 𝑥(𝑈𝑧)
48 nfcv 2751 . . . . . . . . . . . . . . . 16 𝑥(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))
49 nffvmpt1 6111 . . . . . . . . . . . . . . . 16 𝑥((𝑥𝐵𝑋)‘𝑧)
5047, 48, 49nfbr 4629 . . . . . . . . . . . . . . 15 𝑥(𝑈𝑧)(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧)
51 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (𝑈𝑥) = (𝑈𝑧))
52 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ((1st𝐹)‘𝑥) = ((1st𝐹)‘𝑧))
53 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ((1st𝐺)‘𝑥) = ((1st𝐺)‘𝑧))
5452, 53oveq12d 6567 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) = (((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧)))
55 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → ((𝑥𝐵𝑋)‘𝑥) = ((𝑥𝐵𝑋)‘𝑧))
5651, 54, 55breq123d 4597 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → ((𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) ↔ (𝑈𝑧)(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧)))
5750, 56rspc 3276 . . . . . . . . . . . . . 14 (𝑧𝐵 → (∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) → (𝑈𝑧)(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧)))
5839, 46, 57sylc 63 . . . . . . . . . . . . 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 6268 . . . . . . . . . . . . . 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 6268 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐺)‘𝑧) ∈ (Base‘𝐷))
64 eqid 2610 . . . . . . . . . . . . . 14 (Sect‘𝐷) = (Sect‘𝐷)
653, 4, 59, 61, 63, 64isinv 16243 . . . . . . . . . . . . 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 221 . . . . . . . . . . . 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 2610 . . . . . . . . . . . 12 (comp‘𝐷) = (comp‘𝐷)
69 eqid 2610 . . . . . . . . . . . 12 (Id‘𝐷) = (Id‘𝐷)
703, 21, 68, 69, 64, 59, 61, 63issect 16236 . . . . . . . . . . 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 221 . . . . . . . . . 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 1068 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(𝑈𝑧)) = ((Id‘𝐷)‘((1st𝐹)‘𝑧)))
7372oveq1d 6564 . . . . . . . 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 1060 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦𝐵)
7560, 74ffvelrnd 6268 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
76 eqid 2610 . . . . . . . . . . 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 16351 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑦(2nd𝐹)𝑧):(𝑦(Hom ‘𝐶)𝑧)⟶(((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑧)))
79 simpr3 1062 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))
8078, 79ffvelrnd 6268 . . . . . . . . 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 16167 . . . . . . . 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 2645 . . . . . . 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 16434 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑈 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
8683, 85, 10, 21, 39natcl 16436 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑧) ∈ (((1st𝐹)‘𝑧)(Hom ‘𝐷)((1st𝐺)‘𝑧)))
8771simp2d 1067 . . . . . . . 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 16170 . . . . . . 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 16438 . . . . . . . 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 6565 . . . . . . 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 2648 . . . . . 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 6564 . . . . 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 6268 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐺)‘𝑦) ∈ (Base‘𝐷))
94 nfcv 2751 . . . . . . . . . . . . 13 𝑥(𝑈𝑦)
95 nfcv 2751 . . . . . . . . . . . . 13 𝑥(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))
96 nffvmpt1 6111 . . . . . . . . . . . . 13 𝑥((𝑥𝐵𝑋)‘𝑦)
9794, 95, 96nfbr 4629 . . . . . . . . . . . 12 𝑥(𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)
98 fveq2 6103 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑈𝑥) = (𝑈𝑦))
9935, 34oveq12d 6567 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) = (((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦)))
100 fveq2 6103 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑥𝐵𝑋)‘𝑥) = ((𝑥𝐵𝑋)‘𝑦))
10198, 99, 100breq123d 4597 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) ↔ (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)))
10297, 101rspc 3276 . . . . . . . . . . 11 (𝑦𝐵 → (∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) → (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)))
10374, 46, 102sylc 63 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦))
1043, 4, 59, 75, 93, 64isinv 16243 . . . . . . . . . 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 221 . . . . . . . . 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 16236 . . . . . . . 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 221 . . . . . . 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 1066 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥𝐵𝑋)‘𝑦) ∈ (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
110108simp2d 1067 . . . . . . 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 16351 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑦(2nd𝐺)𝑧):(𝑦(Hom ‘𝐶)𝑧)⟶(((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑧)))
113112, 79ffvelrnd 6268 . . . . . . 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 16169 . . . . . 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 16170 . . . . 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 16436 . . . . . . . 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 16170 . . . . . . 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 1068 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈𝑦)(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)) = ((Id‘𝐷)‘((1st𝐺)‘𝑦)))
119118oveq2d 6565 . . . . . . 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 16168 . . . . . . 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 2648 . . . . . 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 6565 . . . . 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 2649 . . . 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 2955 . . 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 16431 . . 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 959 . 2 (𝜑 → (𝑥𝐵𝑋) ∈ (𝐺𝑁𝐹))
127 nfv 1830 . . . 4 𝑦(𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥)
128127, 97, 101cbvral 3143 . . 3 (∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) ↔ ∀𝑦𝐵 (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦))
12945, 128sylib 207 . 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 16456 . 2 (𝜑 → (𝑈(𝐹𝐼𝐺)(𝑥𝐵𝑋) ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ (𝑥𝐵𝑋) ∈ (𝐺𝑁𝐹) ∧ ∀𝑦𝐵 (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦))))
1331, 126, 129, 132mpbir3and 1238 1 (𝜑𝑈(𝐹𝐼𝐺)(𝑥𝐵𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  Rel wrel 5043  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Xcixp 7794  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Idccid 16149  Sectcsect 16227  Invcinv 16228   Func cfunc 16337   Nat cnat 16424   FuncCat cfuc 16425 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-hom 15793  df-cco 15794  df-cat 16152  df-cid 16153  df-sect 16230  df-inv 16231  df-func 16341  df-nat 16426  df-fuc 16427 This theorem is referenced by:  fuciso  16458  yonedainv  16744
 Copyright terms: Public domain W3C validator