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Theorem yonedalem3 16967
 Description: Lemma for yoneda 16970. (Contributed by Mario Carneiro, 28-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))))
Assertion
Ref Expression
yonedalem3 (𝜑𝑀 ∈ (𝑍((𝑄 ×c 𝑂) Nat 𝑇)𝐸))
Distinct variable groups:   𝑓,𝑎,𝑥, 1   𝐶,𝑎,𝑓,𝑥   𝐸,𝑎,𝑓   𝐵,𝑎,𝑓,𝑥   𝑂,𝑎,𝑓,𝑥   𝑆,𝑎,𝑓,𝑥   𝑄,𝑎,𝑓,𝑥   𝑇,𝑓   𝜑,𝑎,𝑓,𝑥   𝑌,𝑎,𝑓,𝑥   𝑍,𝑎,𝑓,𝑥
Allowed substitution hints:   𝑅(𝑥,𝑓,𝑎)   𝑇(𝑥,𝑎)   𝑈(𝑥,𝑓,𝑎)   𝐸(𝑥)   𝐻(𝑥,𝑓,𝑎)   𝑀(𝑥,𝑓,𝑎)   𝑉(𝑥,𝑓,𝑎)   𝑊(𝑥,𝑓,𝑎)

Proof of Theorem yonedalem3
Dummy variables 𝑔 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 yoneda.m . . . . 5 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))
2 ovex 6718 . . . . . 6 (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ∈ V
32mptex 6527 . . . . 5 (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))) ∈ V
41, 3fnmpt2i 7284 . . . 4 𝑀 Fn ((𝑂 Func 𝑆) × 𝐵)
54a1i 11 . . 3 (𝜑𝑀 Fn ((𝑂 Func 𝑆) × 𝐵))
6 yoneda.y . . . . . . . 8 𝑌 = (Yon‘𝐶)
7 yoneda.b . . . . . . . 8 𝐵 = (Base‘𝐶)
8 yoneda.1 . . . . . . . 8 1 = (Id‘𝐶)
9 yoneda.o . . . . . . . 8 𝑂 = (oppCat‘𝐶)
10 yoneda.s . . . . . . . 8 𝑆 = (SetCat‘𝑈)
11 yoneda.t . . . . . . . 8 𝑇 = (SetCat‘𝑉)
12 yoneda.q . . . . . . . 8 𝑄 = (𝑂 FuncCat 𝑆)
13 yoneda.h . . . . . . . 8 𝐻 = (HomF𝑄)
14 yoneda.r . . . . . . . 8 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
15 yoneda.e . . . . . . . 8 𝐸 = (𝑂 evalF 𝑆)
16 yoneda.z . . . . . . . 8 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
17 yoneda.c . . . . . . . . 9 (𝜑𝐶 ∈ Cat)
1817adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → 𝐶 ∈ Cat)
19 yoneda.w . . . . . . . . 9 (𝜑𝑉𝑊)
2019adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → 𝑉𝑊)
21 yoneda.u . . . . . . . . 9 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
2221adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → ran (Homf𝐶) ⊆ 𝑈)
23 yoneda.v . . . . . . . . 9 (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
2423adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
25 simprl 809 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → 𝑔 ∈ (𝑂 Func 𝑆))
26 simprr 811 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → 𝑦𝐵)
276, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 25, 26, 1yonedalem3a 16961 . . . . . . 7 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → ((𝑔𝑀𝑦) = (𝑎 ∈ (((1st𝑌)‘𝑦)(𝑂 Nat 𝑆)𝑔) ↦ ((𝑎𝑦)‘( 1𝑦))) ∧ (𝑔𝑀𝑦):(𝑔(1st𝑍)𝑦)⟶(𝑔(1st𝐸)𝑦)))
2827simprd 478 . . . . . 6 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔𝑀𝑦):(𝑔(1st𝑍)𝑦)⟶(𝑔(1st𝐸)𝑦))
29 eqid 2651 . . . . . . 7 (Hom ‘𝑇) = (Hom ‘𝑇)
30 eqid 2651 . . . . . . . . . . 11 (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
3112fucbas 16667 . . . . . . . . . . 11 (𝑂 Func 𝑆) = (Base‘𝑄)
329, 7oppcbas 16425 . . . . . . . . . . 11 𝐵 = (Base‘𝑂)
3330, 31, 32xpcbas 16865 . . . . . . . . . 10 ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
34 eqid 2651 . . . . . . . . . 10 (Base‘𝑇) = (Base‘𝑇)
35 relfunc 16569 . . . . . . . . . . 11 Rel ((𝑄 ×c 𝑂) Func 𝑇)
366, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 23yonedalem1 16959 . . . . . . . . . . . 12 (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
3736simpld 474 . . . . . . . . . . 11 (𝜑𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
38 1st2ndbr 7261 . . . . . . . . . . 11 ((Rel ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝑍))
3935, 37, 38sylancr 696 . . . . . . . . . 10 (𝜑 → (1st𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝑍))
4033, 34, 39funcf1 16573 . . . . . . . . 9 (𝜑 → (1st𝑍):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
4140fovrnda 6847 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔(1st𝑍)𝑦) ∈ (Base‘𝑇))
4211, 20setcbas 16775 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → 𝑉 = (Base‘𝑇))
4341, 42eleqtrrd 2733 . . . . . . 7 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔(1st𝑍)𝑦) ∈ 𝑉)
4436simprd 478 . . . . . . . . . . 11 (𝜑𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
45 1st2ndbr 7261 . . . . . . . . . . 11 ((Rel ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝐸))
4635, 44, 45sylancr 696 . . . . . . . . . 10 (𝜑 → (1st𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝐸))
4733, 34, 46funcf1 16573 . . . . . . . . 9 (𝜑 → (1st𝐸):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
4847fovrnda 6847 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔(1st𝐸)𝑦) ∈ (Base‘𝑇))
4948, 42eleqtrrd 2733 . . . . . . 7 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔(1st𝐸)𝑦) ∈ 𝑉)
5011, 20, 29, 43, 49elsetchom 16778 . . . . . 6 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → ((𝑔𝑀𝑦) ∈ ((𝑔(1st𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st𝐸)𝑦)) ↔ (𝑔𝑀𝑦):(𝑔(1st𝑍)𝑦)⟶(𝑔(1st𝐸)𝑦)))
5128, 50mpbird 247 . . . . 5 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔𝑀𝑦) ∈ ((𝑔(1st𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st𝐸)𝑦)))
5251ralrimivva 3000 . . . 4 (𝜑 → ∀𝑔 ∈ (𝑂 Func 𝑆)∀𝑦𝐵 (𝑔𝑀𝑦) ∈ ((𝑔(1st𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st𝐸)𝑦)))
53 fveq2 6229 . . . . . . 7 (𝑧 = ⟨𝑔, 𝑦⟩ → (𝑀𝑧) = (𝑀‘⟨𝑔, 𝑦⟩))
54 df-ov 6693 . . . . . . 7 (𝑔𝑀𝑦) = (𝑀‘⟨𝑔, 𝑦⟩)
5553, 54syl6eqr 2703 . . . . . 6 (𝑧 = ⟨𝑔, 𝑦⟩ → (𝑀𝑧) = (𝑔𝑀𝑦))
56 fveq2 6229 . . . . . . . 8 (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st𝑍)‘𝑧) = ((1st𝑍)‘⟨𝑔, 𝑦⟩))
57 df-ov 6693 . . . . . . . 8 (𝑔(1st𝑍)𝑦) = ((1st𝑍)‘⟨𝑔, 𝑦⟩)
5856, 57syl6eqr 2703 . . . . . . 7 (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st𝑍)‘𝑧) = (𝑔(1st𝑍)𝑦))
59 fveq2 6229 . . . . . . . 8 (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st𝐸)‘𝑧) = ((1st𝐸)‘⟨𝑔, 𝑦⟩))
60 df-ov 6693 . . . . . . . 8 (𝑔(1st𝐸)𝑦) = ((1st𝐸)‘⟨𝑔, 𝑦⟩)
6159, 60syl6eqr 2703 . . . . . . 7 (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st𝐸)‘𝑧) = (𝑔(1st𝐸)𝑦))
6258, 61oveq12d 6708 . . . . . 6 (𝑧 = ⟨𝑔, 𝑦⟩ → (((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧)) = ((𝑔(1st𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st𝐸)𝑦)))
6355, 62eleq12d 2724 . . . . 5 (𝑧 = ⟨𝑔, 𝑦⟩ → ((𝑀𝑧) ∈ (((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧)) ↔ (𝑔𝑀𝑦) ∈ ((𝑔(1st𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st𝐸)𝑦))))
6463ralxp 5296 . . . 4 (∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑀𝑧) ∈ (((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧)) ↔ ∀𝑔 ∈ (𝑂 Func 𝑆)∀𝑦𝐵 (𝑔𝑀𝑦) ∈ ((𝑔(1st𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st𝐸)𝑦)))
6552, 64sylibr 224 . . 3 (𝜑 → ∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑀𝑧) ∈ (((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧)))
66 ovex 6718 . . . . . 6 (𝑂 Func 𝑆) ∈ V
67 fvex 6239 . . . . . . 7 (Base‘𝐶) ∈ V
687, 67eqeltri 2726 . . . . . 6 𝐵 ∈ V
6966, 68mpt2ex 7292 . . . . 5 (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥)))) ∈ V
701, 69eqeltri 2726 . . . 4 𝑀 ∈ V
7170elixp 7957 . . 3 (𝑀X𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧)) ↔ (𝑀 Fn ((𝑂 Func 𝑆) × 𝐵) ∧ ∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑀𝑧) ∈ (((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧))))
725, 65, 71sylanbrc 699 . 2 (𝜑𝑀X𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧)))
7317adantr 480 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝐶 ∈ Cat)
7419adantr 480 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑉𝑊)
7521adantr 480 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ran (Homf𝐶) ⊆ 𝑈)
7623adantr 480 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
77 simpr1 1087 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵))
78 xp1st 7242 . . . . . 6 (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) → (1st𝑧) ∈ (𝑂 Func 𝑆))
7977, 78syl 17 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (1st𝑧) ∈ (𝑂 Func 𝑆))
80 xp2nd 7243 . . . . . 6 (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) → (2nd𝑧) ∈ 𝐵)
8177, 80syl 17 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (2nd𝑧) ∈ 𝐵)
82 simpr2 1088 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵))
83 xp1st 7242 . . . . . 6 (𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) → (1st𝑤) ∈ (𝑂 Func 𝑆))
8482, 83syl 17 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (1st𝑤) ∈ (𝑂 Func 𝑆))
85 xp2nd 7243 . . . . . 6 (𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) → (2nd𝑤) ∈ 𝐵)
8682, 85syl 17 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (2nd𝑤) ∈ 𝐵)
87 simpr3 1089 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))
88 eqid 2651 . . . . . . . . . 10 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
8912, 88fuchom 16668 . . . . . . . . 9 (𝑂 Nat 𝑆) = (Hom ‘𝑄)
90 eqid 2651 . . . . . . . . 9 (Hom ‘𝑂) = (Hom ‘𝑂)
91 eqid 2651 . . . . . . . . 9 (Hom ‘(𝑄 ×c 𝑂)) = (Hom ‘(𝑄 ×c 𝑂))
9230, 33, 89, 90, 91, 77, 82xpchom 16867 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤) = (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑧)(Hom ‘𝑂)(2nd𝑤))))
93 eqid 2651 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
9493, 9oppchom 16422 . . . . . . . . 9 ((2nd𝑧)(Hom ‘𝑂)(2nd𝑤)) = ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧))
9594xpeq2i 5170 . . . . . . . 8 (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑧)(Hom ‘𝑂)(2nd𝑤))) = (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧)))
9692, 95syl6eq 2701 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤) = (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧))))
9787, 96eleqtrd 2732 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑔 ∈ (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧))))
98 xp1st 7242 . . . . . 6 (𝑔 ∈ (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧))) → (1st𝑔) ∈ ((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)))
9997, 98syl 17 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (1st𝑔) ∈ ((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)))
100 xp2nd 7243 . . . . . 6 (𝑔 ∈ (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧))) → (2nd𝑔) ∈ ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧)))
10197, 100syl 17 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (2nd𝑔) ∈ ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧)))
1026, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 73, 74, 75, 76, 79, 81, 84, 86, 99, 101, 1yonedalem3b 16966 . . . 4 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (((1st𝑤)𝑀(2nd𝑤))(⟨((1st𝑧)(1st𝑍)(2nd𝑧)), ((1st𝑤)(1st𝑍)(2nd𝑤))⟩(comp‘𝑇)((1st𝑤)(1st𝐸)(2nd𝑤)))((1st𝑔)(⟨(1st𝑧), (2nd𝑧)⟩(2nd𝑍)⟨(1st𝑤), (2nd𝑤)⟩)(2nd𝑔))) = (((1st𝑔)(⟨(1st𝑧), (2nd𝑧)⟩(2nd𝐸)⟨(1st𝑤), (2nd𝑤)⟩)(2nd𝑔))(⟨((1st𝑧)(1st𝑍)(2nd𝑧)), ((1st𝑧)(1st𝐸)(2nd𝑧))⟩(comp‘𝑇)((1st𝑤)(1st𝐸)(2nd𝑤)))((1st𝑧)𝑀(2nd𝑧))))
103 1st2nd2 7249 . . . . . . . . . 10 (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
10477, 103syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
105104fveq2d 6233 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝑍)‘𝑧) = ((1st𝑍)‘⟨(1st𝑧), (2nd𝑧)⟩))
106 df-ov 6693 . . . . . . . 8 ((1st𝑧)(1st𝑍)(2nd𝑧)) = ((1st𝑍)‘⟨(1st𝑧), (2nd𝑧)⟩)
107105, 106syl6eqr 2703 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝑍)‘𝑧) = ((1st𝑧)(1st𝑍)(2nd𝑧)))
108 1st2nd2 7249 . . . . . . . . . 10 (𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
10982, 108syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
110109fveq2d 6233 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝑍)‘𝑤) = ((1st𝑍)‘⟨(1st𝑤), (2nd𝑤)⟩))
111 df-ov 6693 . . . . . . . 8 ((1st𝑤)(1st𝑍)(2nd𝑤)) = ((1st𝑍)‘⟨(1st𝑤), (2nd𝑤)⟩)
112110, 111syl6eqr 2703 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝑍)‘𝑤) = ((1st𝑤)(1st𝑍)(2nd𝑤)))
113107, 112opeq12d 4441 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ⟨((1st𝑍)‘𝑧), ((1st𝑍)‘𝑤)⟩ = ⟨((1st𝑧)(1st𝑍)(2nd𝑧)), ((1st𝑤)(1st𝑍)(2nd𝑤))⟩)
114109fveq2d 6233 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝐸)‘𝑤) = ((1st𝐸)‘⟨(1st𝑤), (2nd𝑤)⟩))
115 df-ov 6693 . . . . . . 7 ((1st𝑤)(1st𝐸)(2nd𝑤)) = ((1st𝐸)‘⟨(1st𝑤), (2nd𝑤)⟩)
116114, 115syl6eqr 2703 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝐸)‘𝑤) = ((1st𝑤)(1st𝐸)(2nd𝑤)))
117113, 116oveq12d 6708 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (⟨((1st𝑍)‘𝑧), ((1st𝑍)‘𝑤)⟩(comp‘𝑇)((1st𝐸)‘𝑤)) = (⟨((1st𝑧)(1st𝑍)(2nd𝑧)), ((1st𝑤)(1st𝑍)(2nd𝑤))⟩(comp‘𝑇)((1st𝑤)(1st𝐸)(2nd𝑤))))
118109fveq2d 6233 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑀𝑤) = (𝑀‘⟨(1st𝑤), (2nd𝑤)⟩))
119 df-ov 6693 . . . . . 6 ((1st𝑤)𝑀(2nd𝑤)) = (𝑀‘⟨(1st𝑤), (2nd𝑤)⟩)
120118, 119syl6eqr 2703 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑀𝑤) = ((1st𝑤)𝑀(2nd𝑤)))
121104, 109oveq12d 6708 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑧(2nd𝑍)𝑤) = (⟨(1st𝑧), (2nd𝑧)⟩(2nd𝑍)⟨(1st𝑤), (2nd𝑤)⟩))
122 1st2nd2 7249 . . . . . . . 8 (𝑔 ∈ (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧))) → 𝑔 = ⟨(1st𝑔), (2nd𝑔)⟩)
12397, 122syl 17 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑔 = ⟨(1st𝑔), (2nd𝑔)⟩)
124121, 123fveq12d 6235 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((𝑧(2nd𝑍)𝑤)‘𝑔) = ((⟨(1st𝑧), (2nd𝑧)⟩(2nd𝑍)⟨(1st𝑤), (2nd𝑤)⟩)‘⟨(1st𝑔), (2nd𝑔)⟩))
125 df-ov 6693 . . . . . 6 ((1st𝑔)(⟨(1st𝑧), (2nd𝑧)⟩(2nd𝑍)⟨(1st𝑤), (2nd𝑤)⟩)(2nd𝑔)) = ((⟨(1st𝑧), (2nd𝑧)⟩(2nd𝑍)⟨(1st𝑤), (2nd𝑤)⟩)‘⟨(1st𝑔), (2nd𝑔)⟩)
126124, 125syl6eqr 2703 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((𝑧(2nd𝑍)𝑤)‘𝑔) = ((1st𝑔)(⟨(1st𝑧), (2nd𝑧)⟩(2nd𝑍)⟨(1st𝑤), (2nd𝑤)⟩)(2nd𝑔)))
127117, 120, 126oveq123d 6711 . . . 4 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((𝑀𝑤)(⟨((1st𝑍)‘𝑧), ((1st𝑍)‘𝑤)⟩(comp‘𝑇)((1st𝐸)‘𝑤))((𝑧(2nd𝑍)𝑤)‘𝑔)) = (((1st𝑤)𝑀(2nd𝑤))(⟨((1st𝑧)(1st𝑍)(2nd𝑧)), ((1st𝑤)(1st𝑍)(2nd𝑤))⟩(comp‘𝑇)((1st𝑤)(1st𝐸)(2nd𝑤)))((1st𝑔)(⟨(1st𝑧), (2nd𝑧)⟩(2nd𝑍)⟨(1st𝑤), (2nd𝑤)⟩)(2nd𝑔))))
128104fveq2d 6233 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝐸)‘𝑧) = ((1st𝐸)‘⟨(1st𝑧), (2nd𝑧)⟩))
129 df-ov 6693 . . . . . . . 8 ((1st𝑧)(1st𝐸)(2nd𝑧)) = ((1st𝐸)‘⟨(1st𝑧), (2nd𝑧)⟩)
130128, 129syl6eqr 2703 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝐸)‘𝑧) = ((1st𝑧)(1st𝐸)(2nd𝑧)))
131107, 130opeq12d 4441 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ⟨((1st𝑍)‘𝑧), ((1st𝐸)‘𝑧)⟩ = ⟨((1st𝑧)(1st𝑍)(2nd𝑧)), ((1st𝑧)(1st𝐸)(2nd𝑧))⟩)
132131, 116oveq12d 6708 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (⟨((1st𝑍)‘𝑧), ((1st𝐸)‘𝑧)⟩(comp‘𝑇)((1st𝐸)‘𝑤)) = (⟨((1st𝑧)(1st𝑍)(2nd𝑧)), ((1st𝑧)(1st𝐸)(2nd𝑧))⟩(comp‘𝑇)((1st𝑤)(1st𝐸)(2nd𝑤))))
133104, 109oveq12d 6708 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑧(2nd𝐸)𝑤) = (⟨(1st𝑧), (2nd𝑧)⟩(2nd𝐸)⟨(1st𝑤), (2nd𝑤)⟩))
134133, 123fveq12d 6235 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((𝑧(2nd𝐸)𝑤)‘𝑔) = ((⟨(1st𝑧), (2nd𝑧)⟩(2nd𝐸)⟨(1st𝑤), (2nd𝑤)⟩)‘⟨(1st𝑔), (2nd𝑔)⟩))
135 df-ov 6693 . . . . . 6 ((1st𝑔)(⟨(1st𝑧), (2nd𝑧)⟩(2nd𝐸)⟨(1st𝑤), (2nd𝑤)⟩)(2nd𝑔)) = ((⟨(1st𝑧), (2nd𝑧)⟩(2nd𝐸)⟨(1st𝑤), (2nd𝑤)⟩)‘⟨(1st𝑔), (2nd𝑔)⟩)
136134, 135syl6eqr 2703 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((𝑧(2nd𝐸)𝑤)‘𝑔) = ((1st𝑔)(⟨(1st𝑧), (2nd𝑧)⟩(2nd𝐸)⟨(1st𝑤), (2nd𝑤)⟩)(2nd𝑔)))
137104fveq2d 6233 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑀𝑧) = (𝑀‘⟨(1st𝑧), (2nd𝑧)⟩))
138 df-ov 6693 . . . . . 6 ((1st𝑧)𝑀(2nd𝑧)) = (𝑀‘⟨(1st𝑧), (2nd𝑧)⟩)
139137, 138syl6eqr 2703 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑀𝑧) = ((1st𝑧)𝑀(2nd𝑧)))
140132, 136, 139oveq123d 6711 . . . 4 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (((𝑧(2nd𝐸)𝑤)‘𝑔)(⟨((1st𝑍)‘𝑧), ((1st𝐸)‘𝑧)⟩(comp‘𝑇)((1st𝐸)‘𝑤))(𝑀𝑧)) = (((1st𝑔)(⟨(1st𝑧), (2nd𝑧)⟩(2nd𝐸)⟨(1st𝑤), (2nd𝑤)⟩)(2nd𝑔))(⟨((1st𝑧)(1st𝑍)(2nd𝑧)), ((1st𝑧)(1st𝐸)(2nd𝑧))⟩(comp‘𝑇)((1st𝑤)(1st𝐸)(2nd𝑤)))((1st𝑧)𝑀(2nd𝑧))))
141102, 127, 1403eqtr4d 2695 . . 3 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((𝑀𝑤)(⟨((1st𝑍)‘𝑧), ((1st𝑍)‘𝑤)⟩(comp‘𝑇)((1st𝐸)‘𝑤))((𝑧(2nd𝑍)𝑤)‘𝑔)) = (((𝑧(2nd𝐸)𝑤)‘𝑔)(⟨((1st𝑍)‘𝑧), ((1st𝐸)‘𝑧)⟩(comp‘𝑇)((1st𝐸)‘𝑤))(𝑀𝑧)))
142141ralrimivvva 3001 . 2 (𝜑 → ∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)∀𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵)∀𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤)((𝑀𝑤)(⟨((1st𝑍)‘𝑧), ((1st𝑍)‘𝑤)⟩(comp‘𝑇)((1st𝐸)‘𝑤))((𝑧(2nd𝑍)𝑤)‘𝑔)) = (((𝑧(2nd𝐸)𝑤)‘𝑔)(⟨((1st𝑍)‘𝑧), ((1st𝐸)‘𝑧)⟩(comp‘𝑇)((1st𝐸)‘𝑤))(𝑀𝑧)))
143 eqid 2651 . . 3 ((𝑄 ×c 𝑂) Nat 𝑇) = ((𝑄 ×c 𝑂) Nat 𝑇)
144 eqid 2651 . . 3 (comp‘𝑇) = (comp‘𝑇)
145143, 33, 91, 29, 144, 37, 44isnat2 16655 . 2 (𝜑 → (𝑀 ∈ (𝑍((𝑄 ×c 𝑂) Nat 𝑇)𝐸) ↔ (𝑀X𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧)) ∧ ∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)∀𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵)∀𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤)((𝑀𝑤)(⟨((1st𝑍)‘𝑧), ((1st𝑍)‘𝑤)⟩(comp‘𝑇)((1st𝐸)‘𝑤))((𝑧(2nd𝑍)𝑤)‘𝑔)) = (((𝑧(2nd𝐸)𝑤)‘𝑔)(⟨((1st𝑍)‘𝑧), ((1st𝐸)‘𝑧)⟩(comp‘𝑇)((1st𝐸)‘𝑤))(𝑀𝑧)))))
14672, 142, 145mpbir2and 977 1 (𝜑𝑀 ∈ (𝑍((𝑄 ×c 𝑂) Nat 𝑇)𝐸))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941  Vcvv 3231   ∪ cun 3605   ⊆ wss 3607  ⟨cop 4216   class class class wbr 4685   ↦ cmpt 4762   × cxp 5141  ran crn 5144  Rel wrel 5148   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   ↦ cmpt2 6692  1st c1st 7208  2nd c2nd 7209  tpos ctpos 7396  Xcixp 7950  Basecbs 15904  Hom chom 15999  compcco 16000  Catccat 16372  Idccid 16373  Homf chomf 16374  oppCatcoppc 16418   Func cfunc 16561   ∘func ccofu 16563   Nat cnat 16648   FuncCat cfuc 16649  SetCatcsetc 16772   ×c cxpc 16855   1stF c1stf 16856   2ndF c2ndf 16857   ⟨,⟩F cprf 16858   evalF cevlf 16896  HomFchof 16935  Yoncyon 16936 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-tpos 7397  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  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-sets 15911  df-ress 15912  df-hom 16013  df-cco 16014  df-cat 16376  df-cid 16377  df-homf 16378  df-comf 16379  df-oppc 16419  df-ssc 16517  df-resc 16518  df-subc 16519  df-func 16565  df-cofu 16567  df-nat 16650  df-fuc 16651  df-setc 16773  df-xpc 16859  df-1stf 16860  df-2ndf 16861  df-prf 16862  df-evlf 16900  df-curf 16901  df-hof 16937  df-yon 16938 This theorem is referenced by:  yonedainv  16968
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