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Theorem yonffthlem 17129
Description: Lemma for yonffth 17131. (Contributed by Mario Carneiro, 29-Jan-2017.)
Hypotheses
Ref Expression
yoneda.y 𝑌 = (Yon‘𝐶)
yoneda.b 𝐵 = (Base‘𝐶)
yoneda.1 1 = (Id‘𝐶)
yoneda.o 𝑂 = (oppCat‘𝐶)
yoneda.s 𝑆 = (SetCat‘𝑈)
yoneda.t 𝑇 = (SetCat‘𝑉)
yoneda.q 𝑄 = (𝑂 FuncCat 𝑆)
yoneda.h 𝐻 = (HomF𝑄)
yoneda.r 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
yoneda.e 𝐸 = (𝑂 evalF 𝑆)
yoneda.z 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
yoneda.c (𝜑𝐶 ∈ Cat)
yoneda.w (𝜑𝑉𝑊)
yoneda.u (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
yoneda.v (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
yoneda.m 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))
yonedainv.i 𝐼 = (Inv‘𝑅)
yonedainv.n 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))
Assertion
Ref Expression
yonffthlem (𝜑𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
Distinct variable groups:   𝑓,𝑎,𝑔,𝑥,𝑦, 1   𝑢,𝑎,𝑔,𝑦,𝐶,𝑓,𝑥   𝐸,𝑎,𝑓,𝑔,𝑢,𝑦   𝐵,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑁,𝑎   𝑂,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑆,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑔,𝑀,𝑢,𝑦   𝑄,𝑎,𝑓,𝑔,𝑢,𝑥   𝑇,𝑓,𝑔,𝑢,𝑦   𝜑,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑢,𝑅   𝑌,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑍,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦
Allowed substitution hints:   𝑄(𝑦)   𝑅(𝑥,𝑦,𝑓,𝑔,𝑎)   𝑇(𝑥,𝑎)   𝑈(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)   1 (𝑢)   𝐸(𝑥)   𝐻(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)   𝐼(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)   𝑀(𝑥,𝑓,𝑎)   𝑁(𝑥,𝑦,𝑢,𝑓,𝑔)   𝑉(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)   𝑊(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)

Proof of Theorem yonffthlem
Dummy variables 𝑤 𝑧 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relfunc 16728 . . 3 Rel (𝐶 Func 𝑄)
2 yoneda.y . . . 4 𝑌 = (Yon‘𝐶)
3 yoneda.c . . . 4 (𝜑𝐶 ∈ Cat)
4 yoneda.o . . . 4 𝑂 = (oppCat‘𝐶)
5 yoneda.s . . . 4 𝑆 = (SetCat‘𝑈)
6 yoneda.q . . . 4 𝑄 = (𝑂 FuncCat 𝑆)
7 yoneda.w . . . . 5 (𝜑𝑉𝑊)
8 yoneda.v . . . . . 6 (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
98unssbd 3942 . . . . 5 (𝜑𝑈𝑉)
107, 9ssexd 4940 . . . 4 (𝜑𝑈 ∈ V)
11 yoneda.u . . . 4 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
122, 3, 4, 5, 6, 10, 11yoncl 17109 . . 3 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
13 1st2nd 7366 . . 3 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → 𝑌 = ⟨(1st𝑌), (2nd𝑌)⟩)
141, 12, 13sylancr 575 . 2 (𝜑𝑌 = ⟨(1st𝑌), (2nd𝑌)⟩)
15 1st2ndbr 7369 . . . . 5 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
161, 12, 15sylancr 575 . . . 4 (𝜑 → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
17 fveq2 6333 . . . . . . . . . . 11 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → (𝑁𝑣) = (𝑁‘⟨((1st𝑌)‘𝑤), 𝑧⟩))
18 df-ov 6798 . . . . . . . . . . 11 (((1st𝑌)‘𝑤)𝑁𝑧) = (𝑁‘⟨((1st𝑌)‘𝑤), 𝑧⟩)
1917, 18syl6eqr 2823 . . . . . . . . . 10 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → (𝑁𝑣) = (((1st𝑌)‘𝑤)𝑁𝑧))
20 fveq2 6333 . . . . . . . . . . . 12 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → ((1st𝐸)‘𝑣) = ((1st𝐸)‘⟨((1st𝑌)‘𝑤), 𝑧⟩))
21 df-ov 6798 . . . . . . . . . . . 12 (((1st𝑌)‘𝑤)(1st𝐸)𝑧) = ((1st𝐸)‘⟨((1st𝑌)‘𝑤), 𝑧⟩)
2220, 21syl6eqr 2823 . . . . . . . . . . 11 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → ((1st𝐸)‘𝑣) = (((1st𝑌)‘𝑤)(1st𝐸)𝑧))
23 fveq2 6333 . . . . . . . . . . . 12 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → ((1st𝑍)‘𝑣) = ((1st𝑍)‘⟨((1st𝑌)‘𝑤), 𝑧⟩))
24 df-ov 6798 . . . . . . . . . . . 12 (((1st𝑌)‘𝑤)(1st𝑍)𝑧) = ((1st𝑍)‘⟨((1st𝑌)‘𝑤), 𝑧⟩)
2523, 24syl6eqr 2823 . . . . . . . . . . 11 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → ((1st𝑍)‘𝑣) = (((1st𝑌)‘𝑤)(1st𝑍)𝑧))
2622, 25oveq12d 6813 . . . . . . . . . 10 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → (((1st𝐸)‘𝑣)(Iso‘𝑇)((1st𝑍)‘𝑣)) = ((((1st𝑌)‘𝑤)(1st𝐸)𝑧)(Iso‘𝑇)(((1st𝑌)‘𝑤)(1st𝑍)𝑧)))
2719, 26eleq12d 2844 . . . . . . . . 9 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → ((𝑁𝑣) ∈ (((1st𝐸)‘𝑣)(Iso‘𝑇)((1st𝑍)‘𝑣)) ↔ (((1st𝑌)‘𝑤)𝑁𝑧) ∈ ((((1st𝑌)‘𝑤)(1st𝐸)𝑧)(Iso‘𝑇)(((1st𝑌)‘𝑤)(1st𝑍)𝑧))))
28 yoneda.r . . . . . . . . . . . . . 14 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
2928fucbas 16826 . . . . . . . . . . . . 13 ((𝑄 ×c 𝑂) Func 𝑇) = (Base‘𝑅)
30 yonedainv.i . . . . . . . . . . . . 13 𝐼 = (Inv‘𝑅)
31 yoneda.b . . . . . . . . . . . . . . . . . 18 𝐵 = (Base‘𝐶)
32 yoneda.1 . . . . . . . . . . . . . . . . . 18 1 = (Id‘𝐶)
33 yoneda.t . . . . . . . . . . . . . . . . . 18 𝑇 = (SetCat‘𝑉)
34 yoneda.h . . . . . . . . . . . . . . . . . 18 𝐻 = (HomF𝑄)
35 yoneda.e . . . . . . . . . . . . . . . . . 18 𝐸 = (𝑂 evalF 𝑆)
36 yoneda.z . . . . . . . . . . . . . . . . . 18 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
372, 31, 32, 4, 5, 33, 6, 34, 28, 35, 36, 3, 7, 11, 8yonedalem1 17119 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
3837simpld 482 . . . . . . . . . . . . . . . 16 (𝜑𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
39 funcrcl 16729 . . . . . . . . . . . . . . . 16 (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) → ((𝑄 ×c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat))
4038, 39syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑄 ×c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat))
4140simpld 482 . . . . . . . . . . . . . 14 (𝜑 → (𝑄 ×c 𝑂) ∈ Cat)
4240simprd 483 . . . . . . . . . . . . . 14 (𝜑𝑇 ∈ Cat)
4328, 41, 42fuccat 16836 . . . . . . . . . . . . 13 (𝜑𝑅 ∈ Cat)
4437simprd 483 . . . . . . . . . . . . 13 (𝜑𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
45 eqid 2771 . . . . . . . . . . . . 13 (Iso‘𝑅) = (Iso‘𝑅)
46 yoneda.m . . . . . . . . . . . . . 14 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))
47 yonedainv.n . . . . . . . . . . . . . 14 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))
482, 31, 32, 4, 5, 33, 6, 34, 28, 35, 36, 3, 7, 11, 8, 46, 30, 47yonedainv 17128 . . . . . . . . . . . . 13 (𝜑𝑀(𝑍𝐼𝐸)𝑁)
4929, 30, 43, 38, 44, 45, 48inviso2 16633 . . . . . . . . . . . 12 (𝜑𝑁 ∈ (𝐸(Iso‘𝑅)𝑍))
50 eqid 2771 . . . . . . . . . . . . . 14 (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
516fucbas 16826 . . . . . . . . . . . . . 14 (𝑂 Func 𝑆) = (Base‘𝑄)
524, 31oppcbas 16584 . . . . . . . . . . . . . 14 𝐵 = (Base‘𝑂)
5350, 51, 52xpcbas 17025 . . . . . . . . . . . . 13 ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
54 eqid 2771 . . . . . . . . . . . . 13 ((𝑄 ×c 𝑂) Nat 𝑇) = ((𝑄 ×c 𝑂) Nat 𝑇)
55 eqid 2771 . . . . . . . . . . . . 13 (Iso‘𝑇) = (Iso‘𝑇)
5628, 53, 54, 44, 38, 45, 55fuciso 16841 . . . . . . . . . . . 12 (𝜑 → (𝑁 ∈ (𝐸(Iso‘𝑅)𝑍) ↔ (𝑁 ∈ (𝐸((𝑄 ×c 𝑂) Nat 𝑇)𝑍) ∧ ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁𝑣) ∈ (((1st𝐸)‘𝑣)(Iso‘𝑇)((1st𝑍)‘𝑣)))))
5749, 56mpbid 222 . . . . . . . . . . 11 (𝜑 → (𝑁 ∈ (𝐸((𝑄 ×c 𝑂) Nat 𝑇)𝑍) ∧ ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁𝑣) ∈ (((1st𝐸)‘𝑣)(Iso‘𝑇)((1st𝑍)‘𝑣))))
5857simprd 483 . . . . . . . . . 10 (𝜑 → ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁𝑣) ∈ (((1st𝐸)‘𝑣)(Iso‘𝑇)((1st𝑍)‘𝑣)))
5958adantr 466 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁𝑣) ∈ (((1st𝐸)‘𝑣)(Iso‘𝑇)((1st𝑍)‘𝑣)))
6031, 51, 16funcf1 16732 . . . . . . . . . . . 12 (𝜑 → (1st𝑌):𝐵⟶(𝑂 Func 𝑆))
6160adantr 466 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (1st𝑌):𝐵⟶(𝑂 Func 𝑆))
62 simprr 756 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑤𝐵)
6361, 62ffvelrnd 6505 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ((1st𝑌)‘𝑤) ∈ (𝑂 Func 𝑆))
64 simprl 754 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑧𝐵)
65 opelxpi 5287 . . . . . . . . . 10 ((((1st𝑌)‘𝑤) ∈ (𝑂 Func 𝑆) ∧ 𝑧𝐵) → ⟨((1st𝑌)‘𝑤), 𝑧⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
6663, 64, 65syl2anc 573 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ⟨((1st𝑌)‘𝑤), 𝑧⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
6727, 59, 66rspcdva 3466 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)𝑁𝑧) ∈ ((((1st𝑌)‘𝑤)(1st𝐸)𝑧)(Iso‘𝑇)(((1st𝑌)‘𝑤)(1st𝑍)𝑧)))
684oppccat 16588 . . . . . . . . . . . . 13 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
693, 68syl 17 . . . . . . . . . . . 12 (𝜑𝑂 ∈ Cat)
7069adantr 466 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑂 ∈ Cat)
715setccat 16941 . . . . . . . . . . . . 13 (𝑈 ∈ V → 𝑆 ∈ Cat)
7210, 71syl 17 . . . . . . . . . . . 12 (𝜑𝑆 ∈ Cat)
7372adantr 466 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑆 ∈ Cat)
7435, 70, 73, 52, 63, 64evlf1 17067 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)(1st𝐸)𝑧) = ((1st ‘((1st𝑌)‘𝑤))‘𝑧))
753adantr 466 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝐶 ∈ Cat)
76 eqid 2771 . . . . . . . . . . 11 (Hom ‘𝐶) = (Hom ‘𝐶)
772, 31, 75, 62, 76, 64yon11 17111 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ((1st ‘((1st𝑌)‘𝑤))‘𝑧) = (𝑧(Hom ‘𝐶)𝑤))
7874, 77eqtrd 2805 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)(1st𝐸)𝑧) = (𝑧(Hom ‘𝐶)𝑤))
797adantr 466 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑉𝑊)
8011adantr 466 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ran (Homf𝐶) ⊆ 𝑈)
818adantr 466 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
822, 31, 32, 4, 5, 33, 6, 34, 28, 35, 36, 75, 79, 80, 81, 63, 64yonedalem21 17120 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)(1st𝑍)𝑧) = (((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
8378, 82oveq12d 6813 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ((((1st𝑌)‘𝑤)(1st𝐸)𝑧)(Iso‘𝑇)(((1st𝑌)‘𝑤)(1st𝑍)𝑧)) = ((𝑧(Hom ‘𝐶)𝑤)(Iso‘𝑇)(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤))))
8467, 83eleqtrd 2852 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)𝑁𝑧) ∈ ((𝑧(Hom ‘𝐶)𝑤)(Iso‘𝑇)(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤))))
859adantr 466 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑈𝑉)
86 eqid 2771 . . . . . . . . . . . . 13 (Base‘𝑆) = (Base‘𝑆)
87 relfunc 16728 . . . . . . . . . . . . . 14 Rel (𝑂 Func 𝑆)
88 1st2ndbr 7369 . . . . . . . . . . . . . 14 ((Rel (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑤) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑤)))
8987, 63, 88sylancr 575 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (1st ‘((1st𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑤)))
9052, 86, 89funcf1 16732 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (1st ‘((1st𝑌)‘𝑤)):𝐵⟶(Base‘𝑆))
9190, 64ffvelrnd 6505 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ∈ (Base‘𝑆))
925, 10setcbas 16934 . . . . . . . . . . . 12 (𝜑𝑈 = (Base‘𝑆))
9392adantr 466 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑈 = (Base‘𝑆))
9491, 93eleqtrrd 2853 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ∈ 𝑈)
9577, 94eqeltrrd 2851 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧(Hom ‘𝐶)𝑤) ∈ 𝑈)
9685, 95sseldd 3753 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧(Hom ‘𝐶)𝑤) ∈ 𝑉)
97 eqid 2771 . . . . . . . . . 10 (Homf𝑄) = (Homf𝑄)
98 eqid 2771 . . . . . . . . . . 11 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
996, 98fuchom 16827 . . . . . . . . . 10 (𝑂 Nat 𝑆) = (Hom ‘𝑄)
10061, 64ffvelrnd 6505 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ((1st𝑌)‘𝑧) ∈ (𝑂 Func 𝑆))
10197, 51, 99, 100, 63homfval 16558 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑧)(Homf𝑄)((1st𝑌)‘𝑤)) = (((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
1028unssad 3941 . . . . . . . . . . 11 (𝜑 → ran (Homf𝑄) ⊆ 𝑉)
103102adantr 466 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ran (Homf𝑄) ⊆ 𝑉)
10497, 51homffn 16559 . . . . . . . . . . . 12 (Homf𝑄) Fn ((𝑂 Func 𝑆) × (𝑂 Func 𝑆))
105104a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (Homf𝑄) Fn ((𝑂 Func 𝑆) × (𝑂 Func 𝑆)))
106 fnovrn 6959 . . . . . . . . . . 11 (((Homf𝑄) Fn ((𝑂 Func 𝑆) × (𝑂 Func 𝑆)) ∧ ((1st𝑌)‘𝑧) ∈ (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑤) ∈ (𝑂 Func 𝑆)) → (((1st𝑌)‘𝑧)(Homf𝑄)((1st𝑌)‘𝑤)) ∈ ran (Homf𝑄))
107105, 100, 63, 106syl3anc 1476 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑧)(Homf𝑄)((1st𝑌)‘𝑤)) ∈ ran (Homf𝑄))
108103, 107sseldd 3753 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑧)(Homf𝑄)((1st𝑌)‘𝑤)) ∈ 𝑉)
109101, 108eqeltrrd 2851 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)) ∈ 𝑉)
11033, 79, 96, 109, 55setciso 16947 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ((((1st𝑌)‘𝑤)𝑁𝑧) ∈ ((𝑧(Hom ‘𝐶)𝑤)(Iso‘𝑇)(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤))) ↔ (((1st𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤))))
11184, 110mpbid 222 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
11275adantr 466 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝐶 ∈ Cat)
113112adantr 466 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → 𝐶 ∈ Cat)
11464adantr 466 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑧𝐵)
115114adantr 466 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → 𝑧𝐵)
116 simpr 471 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → 𝑦𝐵)
1172, 31, 113, 115, 76, 116yon11 17111 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → ((1st ‘((1st𝑌)‘𝑧))‘𝑦) = (𝑦(Hom ‘𝐶)𝑧))
118117eqcomd 2777 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → (𝑦(Hom ‘𝐶)𝑧) = ((1st ‘((1st𝑌)‘𝑧))‘𝑦))
119113adantr 466 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝐶 ∈ Cat)
12062ad3antrrr 709 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑤𝐵)
121115adantr 466 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑧𝐵)
122 eqid 2771 . . . . . . . . . . . . . . 15 (comp‘𝐶) = (comp‘𝐶)
123116adantr 466 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑦𝐵)
124 simpr 471 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
125 simpllr 760 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → ∈ (𝑧(Hom ‘𝐶)𝑤))
1262, 31, 119, 120, 76, 121, 122, 123, 124, 125yon12 17112 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → (((𝑧(2nd ‘((1st𝑌)‘𝑤))𝑦)‘𝑔)‘) = ((⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔))
1272, 31, 119, 121, 76, 120, 122, 123, 125, 124yon2 17113 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((((𝑧(2nd𝑌)𝑤)‘)‘𝑦)‘𝑔) = ((⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔))
128126, 127eqtr4d 2808 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → (((𝑧(2nd ‘((1st𝑌)‘𝑤))𝑦)‘𝑔)‘) = ((((𝑧(2nd𝑌)𝑤)‘)‘𝑦)‘𝑔))
129118, 128mpteq12dva 4867 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st𝑌)‘𝑤))𝑦)‘𝑔)‘)) = (𝑔 ∈ ((1st ‘((1st𝑌)‘𝑧))‘𝑦) ↦ ((((𝑧(2nd𝑌)𝑤)‘)‘𝑦)‘𝑔)))
13016adantr 466 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
13131, 76, 99, 130, 64, 62funcf2 16734 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧(2nd𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)⟶(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
132131ffvelrnda 6504 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd𝑌)𝑤)‘) ∈ (((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
13398, 132nat1st2nd 16817 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd𝑌)𝑤)‘) ∈ (⟨(1st ‘((1st𝑌)‘𝑧)), (2nd ‘((1st𝑌)‘𝑧))⟩(𝑂 Nat 𝑆)⟨(1st ‘((1st𝑌)‘𝑤)), (2nd ‘((1st𝑌)‘𝑤))⟩))
134133adantr 466 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → ((𝑧(2nd𝑌)𝑤)‘) ∈ (⟨(1st ‘((1st𝑌)‘𝑧)), (2nd ‘((1st𝑌)‘𝑧))⟩(𝑂 Nat 𝑆)⟨(1st ‘((1st𝑌)‘𝑤)), (2nd ‘((1st𝑌)‘𝑤))⟩))
135 eqid 2771 . . . . . . . . . . . . . . 15 (Hom ‘𝑆) = (Hom ‘𝑆)
13698, 134, 52, 135, 116natcl 16819 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → (((𝑧(2nd𝑌)𝑤)‘)‘𝑦) ∈ (((1st ‘((1st𝑌)‘𝑧))‘𝑦)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑤))‘𝑦)))
13710adantr 466 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑈 ∈ V)
138137ad2antrr 705 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → 𝑈 ∈ V)
13960ad2antrr 705 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st𝑌):𝐵⟶(𝑂 Func 𝑆))
140139, 114ffvelrnd 6505 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st𝑌)‘𝑧) ∈ (𝑂 Func 𝑆))
141 1st2ndbr 7369 . . . . . . . . . . . . . . . . . . 19 ((Rel (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑧) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st𝑌)‘𝑧))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑧)))
14287, 140, 141sylancr 575 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘((1st𝑌)‘𝑧))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑧)))
14352, 86, 142funcf1 16732 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘((1st𝑌)‘𝑧)):𝐵⟶(Base‘𝑆))
144143ffvelrnda 6504 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → ((1st ‘((1st𝑌)‘𝑧))‘𝑦) ∈ (Base‘𝑆))
14593ad2antrr 705 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → 𝑈 = (Base‘𝑆))
146144, 145eleqtrrd 2853 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → ((1st ‘((1st𝑌)‘𝑧))‘𝑦) ∈ 𝑈)
14790adantr 466 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘((1st𝑌)‘𝑤)):𝐵⟶(Base‘𝑆))
148147ffvelrnda 6504 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → ((1st ‘((1st𝑌)‘𝑤))‘𝑦) ∈ (Base‘𝑆))
149148, 145eleqtrrd 2853 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → ((1st ‘((1st𝑌)‘𝑤))‘𝑦) ∈ 𝑈)
1505, 138, 135, 146, 149elsetchom 16937 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → ((((𝑧(2nd𝑌)𝑤)‘)‘𝑦) ∈ (((1st ‘((1st𝑌)‘𝑧))‘𝑦)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑤))‘𝑦)) ↔ (((𝑧(2nd𝑌)𝑤)‘)‘𝑦):((1st ‘((1st𝑌)‘𝑧))‘𝑦)⟶((1st ‘((1st𝑌)‘𝑤))‘𝑦)))
151136, 150mpbid 222 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → (((𝑧(2nd𝑌)𝑤)‘)‘𝑦):((1st ‘((1st𝑌)‘𝑧))‘𝑦)⟶((1st ‘((1st𝑌)‘𝑤))‘𝑦))
152151feqmptd 6393 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → (((𝑧(2nd𝑌)𝑤)‘)‘𝑦) = (𝑔 ∈ ((1st ‘((1st𝑌)‘𝑧))‘𝑦) ↦ ((((𝑧(2nd𝑌)𝑤)‘)‘𝑦)‘𝑔)))
153129, 152eqtr4d 2808 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st𝑌)‘𝑤))𝑦)‘𝑔)‘)) = (((𝑧(2nd𝑌)𝑤)‘)‘𝑦))
154153mpteq2dva 4879 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st𝑌)‘𝑤))𝑦)‘𝑔)‘))) = (𝑦𝐵 ↦ (((𝑧(2nd𝑌)𝑤)‘)‘𝑦)))
15579adantr 466 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑉𝑊)
15680adantr 466 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ran (Homf𝐶) ⊆ 𝑈)
15781adantr 466 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
15863adantr 466 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st𝑌)‘𝑤) ∈ (𝑂 Func 𝑆))
15977eleq2d 2836 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ( ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ↔ ∈ (𝑧(Hom ‘𝐶)𝑤)))
160159biimpar 463 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑧))
1612, 31, 32, 4, 5, 33, 6, 34, 28, 35, 36, 112, 155, 156, 157, 158, 114, 47, 160yonedalem4a 17122 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((1st𝑌)‘𝑤)𝑁𝑧)‘) = (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st𝑌)‘𝑤))𝑦)‘𝑔)‘))))
16298, 133, 52natfn 16820 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd𝑌)𝑤)‘) Fn 𝐵)
163 dffn5 6385 . . . . . . . . . . 11 (((𝑧(2nd𝑌)𝑤)‘) Fn 𝐵 ↔ ((𝑧(2nd𝑌)𝑤)‘) = (𝑦𝐵 ↦ (((𝑧(2nd𝑌)𝑤)‘)‘𝑦)))
164162, 163sylib 208 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd𝑌)𝑤)‘) = (𝑦𝐵 ↦ (((𝑧(2nd𝑌)𝑤)‘)‘𝑦)))
165154, 161, 1643eqtr4d 2815 . . . . . . . . 9 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((1st𝑌)‘𝑤)𝑁𝑧)‘) = ((𝑧(2nd𝑌)𝑤)‘))
166165mpteq2dva 4879 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ( ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((((1st𝑌)‘𝑤)𝑁𝑧)‘)) = ( ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((𝑧(2nd𝑌)𝑤)‘)))
167 f1of 6279 . . . . . . . . . 10 ((((1st𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)) → (((1st𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)⟶(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
168111, 167syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)⟶(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
169168feqmptd 6393 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)𝑁𝑧) = ( ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((((1st𝑌)‘𝑤)𝑁𝑧)‘)))
170131feqmptd 6393 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧(2nd𝑌)𝑤) = ( ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((𝑧(2nd𝑌)𝑤)‘)))
171166, 169, 1703eqtr4d 2815 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)𝑁𝑧) = (𝑧(2nd𝑌)𝑤))
172 f1oeq1 6269 . . . . . . 7 ((((1st𝑌)‘𝑤)𝑁𝑧) = (𝑧(2nd𝑌)𝑤) → ((((1st𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)) ↔ (𝑧(2nd𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤))))
173171, 172syl 17 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ((((1st𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)) ↔ (𝑧(2nd𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤))))
174111, 173mpbid 222 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧(2nd𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
175174ralrimivva 3120 . . . 4 (𝜑 → ∀𝑧𝐵𝑤𝐵 (𝑧(2nd𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
17631, 76, 99isffth2 16782 . . . 4 ((1st𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd𝑌) ↔ ((1st𝑌)(𝐶 Func 𝑄)(2nd𝑌) ∧ ∀𝑧𝐵𝑤𝐵 (𝑧(2nd𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤))))
17716, 175, 176sylanbrc 572 . . 3 (𝜑 → (1st𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd𝑌))
178 df-br 4788 . . 3 ((1st𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd𝑌) ↔ ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
179177, 178sylib 208 . 2 (𝜑 → ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
18014, 179eqeltrd 2850 1 (𝜑𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  cun 3721  cin 3722  wss 3723  cop 4323   class class class wbr 4787  cmpt 4864   × cxp 5248  ran crn 5251  Rel wrel 5255   Fn wfn 6025  wf 6026  1-1-ontowf1o 6029  cfv 6030  (class class class)co 6795  cmpt2 6797  1st c1st 7316  2nd c2nd 7317  tpos ctpos 7506  Basecbs 16063  Hom chom 16159  compcco 16160  Catccat 16531  Idccid 16532  Homf chomf 16533  oppCatcoppc 16577  Invcinv 16611  Isociso 16612   Func cfunc 16720  func ccofu 16722   Full cful 16768   Faith cfth 16769   Nat cnat 16807   FuncCat cfuc 16808  SetCatcsetc 16931   ×c cxpc 17015   1stF c1stf 17016   2ndF c2ndf 17017   ⟨,⟩F cprf 17018   evalF cevlf 17056  HomFchof 17095  Yoncyon 17096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099  ax-cnex 10197  ax-resscn 10198  ax-1cn 10199  ax-icn 10200  ax-addcl 10201  ax-addrcl 10202  ax-mulcl 10203  ax-mulrcl 10204  ax-mulcom 10205  ax-addass 10206  ax-mulass 10207  ax-distr 10208  ax-i2m1 10209  ax-1ne0 10210  ax-1rid 10211  ax-rnegex 10212  ax-rrecex 10213  ax-cnre 10214  ax-pre-lttri 10215  ax-pre-lttrn 10216  ax-pre-ltadd 10217  ax-pre-mulgt0 10218
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6756  df-ov 6798  df-oprab 6799  df-mpt2 6800  df-om 7216  df-1st 7318  df-2nd 7319  df-tpos 7507  df-wrecs 7562  df-recs 7624  df-rdg 7662  df-1o 7716  df-oadd 7720  df-er 7899  df-map 8014  df-pm 8015  df-ixp 8066  df-en 8113  df-dom 8114  df-sdom 8115  df-fin 8116  df-pnf 10281  df-mnf 10282  df-xr 10283  df-ltxr 10284  df-le 10285  df-sub 10473  df-neg 10474  df-nn 11226  df-2 11284  df-3 11285  df-4 11286  df-5 11287  df-6 11288  df-7 11289  df-8 11290  df-9 11291  df-n0 11499  df-z 11584  df-dec 11700  df-uz 11893  df-fz 12533  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-ress 16071  df-hom 16173  df-cco 16174  df-cat 16535  df-cid 16536  df-homf 16537  df-comf 16538  df-oppc 16578  df-sect 16613  df-inv 16614  df-iso 16615  df-ssc 16676  df-resc 16677  df-subc 16678  df-func 16724  df-cofu 16726  df-full 16770  df-fth 16771  df-nat 16809  df-fuc 16810  df-setc 16932  df-xpc 17019  df-1stf 17020  df-2ndf 17021  df-prf 17022  df-evlf 17060  df-curf 17061  df-hof 17097  df-yon 17098
This theorem is referenced by:  yonffth  17131
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