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Theorem invfuc 17863
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 2736 . . . . . . . . . 10 (Base‘𝐷) = (Base‘𝐷)
4 fucinv.j . . . . . . . . . 10 𝐽 = (Inv‘𝐷)
5 fuciso.f . . . . . . . . . . . . 13 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
6 funcrcl 17749 . . . . . . . . . . . . 13 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
75, 6syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
87simprd 496 . . . . . . . . . . 11 (𝜑𝐷 ∈ Cat)
98adantr 481 . . . . . . . . . 10 ((𝜑𝑥𝐵) → 𝐷 ∈ Cat)
10 fuciso.b . . . . . . . . . . . 12 𝐵 = (Base‘𝐶)
11 relfunc 17748 . . . . . . . . . . . . 13 Rel (𝐶 Func 𝐷)
12 1st2ndbr 7974 . . . . . . . . . . . . 13 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1311, 5, 12sylancr 587 . . . . . . . . . . . 12 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1410, 3, 13funcf1 17752 . . . . . . . . . . 11 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
1514ffvelcdmda 7035 . . . . . . . . . 10 ((𝜑𝑥𝐵) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
16 fuciso.g . . . . . . . . . . . . 13 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
17 1st2ndbr 7974 . . . . . . . . . . . . 13 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
1811, 16, 17sylancr 587 . . . . . . . . . . . 12 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
1910, 3, 18funcf1 17752 . . . . . . . . . . 11 (𝜑 → (1st𝐺):𝐵⟶(Base‘𝐷))
2019ffvelcdmda 7035 . . . . . . . . . 10 ((𝜑𝑥𝐵) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
21 eqid 2736 . . . . . . . . . 10 (Hom ‘𝐷) = (Hom ‘𝐷)
223, 4, 9, 15, 20, 21invss 17644 . . . . . . . . 9 ((𝜑𝑥𝐵) → (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) ⊆ ((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)) × (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥))))
2322ssbrd 5148 . . . . . . . 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 5681 . . . . . . . 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 497 . . . . . . 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 3143 . . . . 5 (𝜑 → ∀𝑥𝐵 𝑋 ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
2910fvexi 6856 . . . . . 6 𝐵 ∈ V
30 mptelixpg 8873 . . . . . 6 (𝐵 ∈ V → ((𝑥𝐵𝑋) ∈ X𝑥𝐵 (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)) ↔ ∀𝑥𝐵 𝑋 ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥))))
3129, 30ax-mp 5 . . . . 5 ((𝑥𝐵𝑋) ∈ X𝑥𝐵 (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)) ↔ ∀𝑥𝐵 𝑋 ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
3228, 31sylibr 233 . . . 4 (𝜑 → (𝑥𝐵𝑋) ∈ X𝑥𝐵 (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
33 fveq2 6842 . . . . . 6 (𝑥 = 𝑦 → ((1st𝐺)‘𝑥) = ((1st𝐺)‘𝑦))
34 fveq2 6842 . . . . . 6 (𝑥 = 𝑦 → ((1st𝐹)‘𝑥) = ((1st𝐹)‘𝑦))
3533, 34oveq12d 7375 . . . . 5 (𝑥 = 𝑦 → (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)) = (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
3635cbvixpv 8853 . . . 4 X𝑥𝐵 (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)) = X𝑦𝐵 (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦))
3732, 36eleqtrdi 2848 . . 3 (𝜑 → (𝑥𝐵𝑋) ∈ X𝑦𝐵 (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
38 simpr2 1195 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧𝐵)
39 simpr 485 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐵) → 𝑥𝐵)
40 eqid 2736 . . . . . . . . . . . . . . . . . . 19 (𝑥𝐵𝑋) = (𝑥𝐵𝑋)
4140fvmpt2 6959 . . . . . . . . . . . . . . . . . 18 ((𝑥𝐵𝑋 ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥))) → ((𝑥𝐵𝑋)‘𝑥) = 𝑋)
4239, 27, 41syl2anc 584 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐵) → ((𝑥𝐵𝑋)‘𝑥) = 𝑋)
432, 42breqtrrd 5133 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐵) → (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥))
4443ralrimiva 3143 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥))
4544adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥))
46 nfcv 2907 . . . . . . . . . . . . . . . 16 𝑥(𝑈𝑧)
47 nfcv 2907 . . . . . . . . . . . . . . . 16 𝑥(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))
48 nffvmpt1 6853 . . . . . . . . . . . . . . . 16 𝑥((𝑥𝐵𝑋)‘𝑧)
4946, 47, 48nfbr 5152 . . . . . . . . . . . . . . 15 𝑥(𝑈𝑧)(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧)
50 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (𝑈𝑥) = (𝑈𝑧))
51 fveq2 6842 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ((1st𝐹)‘𝑥) = ((1st𝐹)‘𝑧))
52 fveq2 6842 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ((1st𝐺)‘𝑥) = ((1st𝐺)‘𝑧))
5351, 52oveq12d 7375 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) = (((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧)))
54 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → ((𝑥𝐵𝑋)‘𝑥) = ((𝑥𝐵𝑋)‘𝑧))
5550, 53, 54breq123d 5119 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → ((𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) ↔ (𝑈𝑧)(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧)))
5649, 55rspc 3569 . . . . . . . . . . . . . 14 (𝑧𝐵 → (∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) → (𝑈𝑧)(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧)))
5738, 45, 56sylc 65 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑧)(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧))
588adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐷 ∈ Cat)
5914adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐹):𝐵⟶(Base‘𝐷))
6059, 38ffvelcdmd 7036 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐹)‘𝑧) ∈ (Base‘𝐷))
6119adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐺):𝐵⟶(Base‘𝐷))
6261, 38ffvelcdmd 7036 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐺)‘𝑧) ∈ (Base‘𝐷))
63 eqid 2736 . . . . . . . . . . . . . 14 (Sect‘𝐷) = (Sect‘𝐷)
643, 4, 58, 60, 62, 63isinv 17643 . . . . . . . . . . . . 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))))
6557, 64mpbid 231 . . . . . . . . . . . 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)))
6665simpld 495 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑧)(((1st𝐹)‘𝑧)(Sect‘𝐷)((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧))
67 eqid 2736 . . . . . . . . . . . 12 (comp‘𝐷) = (comp‘𝐷)
68 eqid 2736 . . . . . . . . . . . 12 (Id‘𝐷) = (Id‘𝐷)
693, 21, 67, 68, 63, 58, 60, 62issect 17636 . . . . . . . . . . 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)))))
7066, 69mpbid 231 . . . . . . . . . 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))))
7170simp3d 1144 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(𝑈𝑧)) = ((Id‘𝐷)‘((1st𝐹)‘𝑧)))
7271oveq1d 7372 . . . . . . . 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)))
73 simpr1 1194 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦𝐵)
7459, 73ffvelcdmd 7036 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
75 eqid 2736 . . . . . . . . . . 11 (Hom ‘𝐶) = (Hom ‘𝐶)
7613adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
7710, 75, 21, 76, 73, 38funcf2 17754 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑦(2nd𝐹)𝑧):(𝑦(Hom ‘𝐶)𝑧)⟶(((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑧)))
78 simpr3 1196 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))
7977, 78ffvelcdmd 7036 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2nd𝐹)𝑧)‘𝑓) ∈ (((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑧)))
803, 21, 68, 58, 74, 67, 60, 79catlid 17563 . . . . . . . 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))
8172, 80eqtr2d 2777 . . . . . . 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)))
82 fuciso.n . . . . . . . . 9 𝑁 = (𝐶 Nat 𝐷)
831adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑈 ∈ (𝐹𝑁𝐺))
8482, 83nat1st2nd 17838 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑈 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
8582, 84, 10, 21, 38natcl 17840 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑧) ∈ (((1st𝐹)‘𝑧)(Hom ‘𝐷)((1st𝐺)‘𝑧)))
8670simp2d 1143 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥𝐵𝑋)‘𝑧) ∈ (((1st𝐺)‘𝑧)(Hom ‘𝐷)((1st𝐹)‘𝑧)))
873, 21, 67, 58, 74, 60, 62, 79, 85, 60, 86catass 17566 . . . . . . 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))))
8882, 84, 10, 75, 67, 73, 38, 78nati 17842 . . . . . . . 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)))
8988oveq2d 7373 . . . . . . 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))))
9081, 87, 893eqtrd 2780 . . . . . 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))))
9190oveq1d 7372 . . . . 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)))
9261, 73ffvelcdmd 7036 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐺)‘𝑦) ∈ (Base‘𝐷))
93 nfcv 2907 . . . . . . . . . . . . 13 𝑥(𝑈𝑦)
94 nfcv 2907 . . . . . . . . . . . . 13 𝑥(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))
95 nffvmpt1 6853 . . . . . . . . . . . . 13 𝑥((𝑥𝐵𝑋)‘𝑦)
9693, 94, 95nfbr 5152 . . . . . . . . . . . 12 𝑥(𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)
97 fveq2 6842 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑈𝑥) = (𝑈𝑦))
9834, 33oveq12d 7375 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) = (((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦)))
99 fveq2 6842 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑥𝐵𝑋)‘𝑥) = ((𝑥𝐵𝑋)‘𝑦))
10097, 98, 99breq123d 5119 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) ↔ (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)))
10196, 100rspc 3569 . . . . . . . . . . 11 (𝑦𝐵 → (∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) → (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)))
10273, 45, 101sylc 65 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦))
1033, 4, 58, 74, 92, 63isinv 17643 . . . . . . . . . 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))))
104102, 103mpbid 231 . . . . . . . . 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)))
105104simprd 496 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥𝐵𝑋)‘𝑦)(((1st𝐺)‘𝑦)(Sect‘𝐷)((1st𝐹)‘𝑦))(𝑈𝑦))
1063, 21, 67, 68, 63, 58, 92, 74issect 17636 . . . . . . . 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)))))
107105, 106mpbid 231 . . . . . . 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))))
108107simp1d 1142 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥𝐵𝑋)‘𝑦) ∈ (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
109107simp2d 1143 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑦) ∈ (((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑦)))
11018adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
11110, 75, 21, 110, 73, 38funcf2 17754 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑦(2nd𝐺)𝑧):(𝑦(Hom ‘𝐶)𝑧)⟶(((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑧)))
112111, 78ffvelcdmd 7036 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2nd𝐺)𝑧)‘𝑓) ∈ (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑧)))
1133, 21, 67, 58, 74, 92, 62, 109, 112catcocl 17565 . . . . . 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)))
1143, 21, 67, 58, 92, 74, 62, 108, 113, 60, 86catass 17566 . . . . 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))))
11582, 84, 10, 21, 73natcl 17840 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑦) ∈ (((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑦)))
1163, 21, 67, 58, 92, 74, 92, 108, 115, 62, 112catass 17566 . . . . . . 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))))
117107simp3d 1144 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈𝑦)(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)) = ((Id‘𝐷)‘((1st𝐺)‘𝑦)))
118117oveq2d 7373 . . . . . . 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))))
1193, 21, 68, 58, 92, 67, 62, 112catrid 17564 . . . . . . 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))
120116, 118, 1193eqtrd 2780 . . . . . 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))
121120oveq2d 7373 . . . . 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)))
12291, 114, 1213eqtrrd 2781 . . . 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)))
123122ralrimivvva 3200 . . 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)))
12482, 10, 75, 21, 67, 16, 5isnat2 17835 . . 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)))))
12537, 123, 124mpbir2and 711 . 2 (𝜑 → (𝑥𝐵𝑋) ∈ (𝐺𝑁𝐹))
126 nfv 1917 . . . 4 𝑦(𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥)
127126, 96, 100cbvralw 3289 . . 3 (∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) ↔ ∀𝑦𝐵 (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦))
12844, 127sylib 217 . 2 (𝜑 → ∀𝑦𝐵 (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦))
129 fuciso.q . . 3 𝑄 = (𝐶 FuncCat 𝐷)
130 fucinv.i . . 3 𝐼 = (Inv‘𝑄)
131129, 10, 82, 5, 16, 130, 4fucinv 17862 . 2 (𝜑 → (𝑈(𝐹𝐼𝐺)(𝑥𝐵𝑋) ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ (𝑥𝐵𝑋) ∈ (𝐺𝑁𝐹) ∧ ∀𝑦𝐵 (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦))))
1321, 125, 128, 131mpbir3and 1342 1 (𝜑𝑈(𝐹𝐼𝐺)(𝑥𝐵𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  cop 4592   class class class wbr 5105  cmpt 5188   × cxp 5631  Rel wrel 5638  wf 6492  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  Xcixp 8835  Basecbs 17083  Hom chom 17144  compcco 17145  Catccat 17544  Idccid 17545  Sectcsect 17627  Invcinv 17628   Func cfunc 17740   Nat cnat 17828   FuncCat cfuc 17829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-slot 17054  df-ndx 17066  df-base 17084  df-hom 17157  df-cco 17158  df-cat 17548  df-cid 17549  df-sect 17630  df-inv 17631  df-func 17744  df-nat 17830  df-fuc 17831
This theorem is referenced by:  fuciso  17864  yonedainv  18170
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