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Theorem yonedalem3 17518
Description: Lemma for yoneda 17521. (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 7178 . . . . . 6 (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ∈ V
32mptex 6977 . . . . 5 (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))) ∈ V
41, 3fnmpoi 7757 . . . 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 481 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → 𝐶 ∈ Cat)
19 yoneda.w . . . . . . . . 9 (𝜑𝑉𝑊)
2019adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → 𝑉𝑊)
21 yoneda.u . . . . . . . . 9 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
2221adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → ran (Homf𝐶) ⊆ 𝑈)
23 yoneda.v . . . . . . . . 9 (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
2423adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
25 simprl 767 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → 𝑔 ∈ (𝑂 Func 𝑆))
26 simprr 769 . . . . . . . 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 17512 . . . . . . 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 496 . . . . . 6 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔𝑀𝑦):(𝑔(1st𝑍)𝑦)⟶(𝑔(1st𝐸)𝑦))
29 eqid 2818 . . . . . . 7 (Hom ‘𝑇) = (Hom ‘𝑇)
30 eqid 2818 . . . . . . . . . . 11 (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
3112fucbas 17218 . . . . . . . . . . 11 (𝑂 Func 𝑆) = (Base‘𝑄)
329, 7oppcbas 16976 . . . . . . . . . . 11 𝐵 = (Base‘𝑂)
3330, 31, 32xpcbas 17416 . . . . . . . . . 10 ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
34 eqid 2818 . . . . . . . . . 10 (Base‘𝑇) = (Base‘𝑇)
35 relfunc 17120 . . . . . . . . . . 11 Rel ((𝑄 ×c 𝑂) Func 𝑇)
366, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 23yonedalem1 17510 . . . . . . . . . . . 12 (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
3736simpld 495 . . . . . . . . . . 11 (𝜑𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
38 1st2ndbr 7730 . . . . . . . . . . 11 ((Rel ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝑍))
3935, 37, 38sylancr 587 . . . . . . . . . 10 (𝜑 → (1st𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝑍))
4033, 34, 39funcf1 17124 . . . . . . . . 9 (𝜑 → (1st𝑍):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
4140fovrnda 7308 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔(1st𝑍)𝑦) ∈ (Base‘𝑇))
4211, 20setcbas 17326 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → 𝑉 = (Base‘𝑇))
4341, 42eleqtrrd 2913 . . . . . . 7 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔(1st𝑍)𝑦) ∈ 𝑉)
4436simprd 496 . . . . . . . . . . 11 (𝜑𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
45 1st2ndbr 7730 . . . . . . . . . . 11 ((Rel ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝐸))
4635, 44, 45sylancr 587 . . . . . . . . . 10 (𝜑 → (1st𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝐸))
4733, 34, 46funcf1 17124 . . . . . . . . 9 (𝜑 → (1st𝐸):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
4847fovrnda 7308 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔(1st𝐸)𝑦) ∈ (Base‘𝑇))
4948, 42eleqtrrd 2913 . . . . . . 7 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔(1st𝐸)𝑦) ∈ 𝑉)
5011, 20, 29, 43, 49elsetchom 17329 . . . . . 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 258 . . . . 5 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔𝑀𝑦) ∈ ((𝑔(1st𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st𝐸)𝑦)))
5251ralrimivva 3188 . . . 4 (𝜑 → ∀𝑔 ∈ (𝑂 Func 𝑆)∀𝑦𝐵 (𝑔𝑀𝑦) ∈ ((𝑔(1st𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st𝐸)𝑦)))
53 fveq2 6663 . . . . . . 7 (𝑧 = ⟨𝑔, 𝑦⟩ → (𝑀𝑧) = (𝑀‘⟨𝑔, 𝑦⟩))
54 df-ov 7148 . . . . . . 7 (𝑔𝑀𝑦) = (𝑀‘⟨𝑔, 𝑦⟩)
5553, 54syl6eqr 2871 . . . . . 6 (𝑧 = ⟨𝑔, 𝑦⟩ → (𝑀𝑧) = (𝑔𝑀𝑦))
56 fveq2 6663 . . . . . . . 8 (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st𝑍)‘𝑧) = ((1st𝑍)‘⟨𝑔, 𝑦⟩))
57 df-ov 7148 . . . . . . . 8 (𝑔(1st𝑍)𝑦) = ((1st𝑍)‘⟨𝑔, 𝑦⟩)
5856, 57syl6eqr 2871 . . . . . . 7 (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st𝑍)‘𝑧) = (𝑔(1st𝑍)𝑦))
59 fveq2 6663 . . . . . . . 8 (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st𝐸)‘𝑧) = ((1st𝐸)‘⟨𝑔, 𝑦⟩))
60 df-ov 7148 . . . . . . . 8 (𝑔(1st𝐸)𝑦) = ((1st𝐸)‘⟨𝑔, 𝑦⟩)
6159, 60syl6eqr 2871 . . . . . . 7 (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st𝐸)‘𝑧) = (𝑔(1st𝐸)𝑦))
6258, 61oveq12d 7163 . . . . . 6 (𝑧 = ⟨𝑔, 𝑦⟩ → (((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧)) = ((𝑔(1st𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st𝐸)𝑦)))
6355, 62eleq12d 2904 . . . . 5 (𝑧 = ⟨𝑔, 𝑦⟩ → ((𝑀𝑧) ∈ (((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧)) ↔ (𝑔𝑀𝑦) ∈ ((𝑔(1st𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st𝐸)𝑦))))
6463ralxp 5705 . . . 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 235 . . 3 (𝜑 → ∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑀𝑧) ∈ (((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧)))
66 ovex 7178 . . . . . 6 (𝑂 Func 𝑆) ∈ V
677fvexi 6677 . . . . . 6 𝐵 ∈ V
6866, 67mpoex 7766 . . . . 5 (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥)))) ∈ V
691, 68eqeltri 2906 . . . 4 𝑀 ∈ V
7069elixp 8456 . . 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))))
715, 65, 70sylanbrc 583 . 2 (𝜑𝑀X𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧)))
7217adantr 481 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝐶 ∈ Cat)
7319adantr 481 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑉𝑊)
7421adantr 481 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ran (Homf𝐶) ⊆ 𝑈)
7523adantr 481 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
76 simpr1 1186 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵))
77 xp1st 7710 . . . . . 6 (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) → (1st𝑧) ∈ (𝑂 Func 𝑆))
7876, 77syl 17 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (1st𝑧) ∈ (𝑂 Func 𝑆))
79 xp2nd 7711 . . . . . 6 (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) → (2nd𝑧) ∈ 𝐵)
8076, 79syl 17 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (2nd𝑧) ∈ 𝐵)
81 simpr2 1187 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵))
82 xp1st 7710 . . . . . 6 (𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) → (1st𝑤) ∈ (𝑂 Func 𝑆))
8381, 82syl 17 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (1st𝑤) ∈ (𝑂 Func 𝑆))
84 xp2nd 7711 . . . . . 6 (𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) → (2nd𝑤) ∈ 𝐵)
8581, 84syl 17 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (2nd𝑤) ∈ 𝐵)
86 simpr3 1188 . . . . . . 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))
87 eqid 2818 . . . . . . . . . 10 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
8812, 87fuchom 17219 . . . . . . . . 9 (𝑂 Nat 𝑆) = (Hom ‘𝑄)
89 eqid 2818 . . . . . . . . 9 (Hom ‘𝑂) = (Hom ‘𝑂)
90 eqid 2818 . . . . . . . . 9 (Hom ‘(𝑄 ×c 𝑂)) = (Hom ‘(𝑄 ×c 𝑂))
9130, 33, 88, 89, 90, 76, 81xpchom 17418 . . . . . . . 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))))
92 eqid 2818 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
9392, 9oppchom 16973 . . . . . . . . 9 ((2nd𝑧)(Hom ‘𝑂)(2nd𝑤)) = ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧))
9493xpeq2i 5575 . . . . . . . 8 (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑧)(Hom ‘𝑂)(2nd𝑤))) = (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧)))
9591, 94syl6eq 2869 . . . . . . 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))))
9686, 95eleqtrd 2912 . . . . . 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))))
97 xp1st 7710 . . . . . 6 (𝑔 ∈ (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧))) → (1st𝑔) ∈ ((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)))
9896, 97syl 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)))
99 xp2nd 7711 . . . . . 6 (𝑔 ∈ (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧))) → (2nd𝑔) ∈ ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧)))
10096, 99syl 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)))
1016, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 72, 73, 74, 75, 78, 80, 83, 85, 98, 100, 1yonedalem3b 17517 . . . 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))))
102 1st2nd2 7717 . . . . . . . . . 10 (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
10376, 102syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
104103fveq2d 6667 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝑍)‘𝑧) = ((1st𝑍)‘⟨(1st𝑧), (2nd𝑧)⟩))
105 df-ov 7148 . . . . . . . 8 ((1st𝑧)(1st𝑍)(2nd𝑧)) = ((1st𝑍)‘⟨(1st𝑧), (2nd𝑧)⟩)
106104, 105syl6eqr 2871 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝑍)‘𝑧) = ((1st𝑧)(1st𝑍)(2nd𝑧)))
107 1st2nd2 7717 . . . . . . . . . 10 (𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
10881, 107syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
109108fveq2d 6667 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝑍)‘𝑤) = ((1st𝑍)‘⟨(1st𝑤), (2nd𝑤)⟩))
110 df-ov 7148 . . . . . . . 8 ((1st𝑤)(1st𝑍)(2nd𝑤)) = ((1st𝑍)‘⟨(1st𝑤), (2nd𝑤)⟩)
111109, 110syl6eqr 2871 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝑍)‘𝑤) = ((1st𝑤)(1st𝑍)(2nd𝑤)))
112106, 111opeq12d 4803 . . . . . 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))⟩)
113108fveq2d 6667 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝐸)‘𝑤) = ((1st𝐸)‘⟨(1st𝑤), (2nd𝑤)⟩))
114 df-ov 7148 . . . . . . 7 ((1st𝑤)(1st𝐸)(2nd𝑤)) = ((1st𝐸)‘⟨(1st𝑤), (2nd𝑤)⟩)
115113, 114syl6eqr 2871 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝐸)‘𝑤) = ((1st𝑤)(1st𝐸)(2nd𝑤)))
116112, 115oveq12d 7163 . . . . 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))))
117108fveq2d 6667 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑀𝑤) = (𝑀‘⟨(1st𝑤), (2nd𝑤)⟩))
118 df-ov 7148 . . . . . 6 ((1st𝑤)𝑀(2nd𝑤)) = (𝑀‘⟨(1st𝑤), (2nd𝑤)⟩)
119117, 118syl6eqr 2871 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑀𝑤) = ((1st𝑤)𝑀(2nd𝑤)))
120103, 108oveq12d 7163 . . . . . . 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)⟩))
121 1st2nd2 7717 . . . . . . . 8 (𝑔 ∈ (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧))) → 𝑔 = ⟨(1st𝑔), (2nd𝑔)⟩)
12296, 121syl 17 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑔 = ⟨(1st𝑔), (2nd𝑔)⟩)
123120, 122fveq12d 6670 . . . . . 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)⟩))
124 df-ov 7148 . . . . . 6 ((1st𝑔)(⟨(1st𝑧), (2nd𝑧)⟩(2nd𝑍)⟨(1st𝑤), (2nd𝑤)⟩)(2nd𝑔)) = ((⟨(1st𝑧), (2nd𝑧)⟩(2nd𝑍)⟨(1st𝑤), (2nd𝑤)⟩)‘⟨(1st𝑔), (2nd𝑔)⟩)
125123, 124syl6eqr 2871 . . . . 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)))
126116, 119, 125oveq123d 7166 . . . 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))))
127103fveq2d 6667 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝐸)‘𝑧) = ((1st𝐸)‘⟨(1st𝑧), (2nd𝑧)⟩))
128 df-ov 7148 . . . . . . . 8 ((1st𝑧)(1st𝐸)(2nd𝑧)) = ((1st𝐸)‘⟨(1st𝑧), (2nd𝑧)⟩)
129127, 128syl6eqr 2871 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝐸)‘𝑧) = ((1st𝑧)(1st𝐸)(2nd𝑧)))
130106, 129opeq12d 4803 . . . . . 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))⟩)
131130, 115oveq12d 7163 . . . . 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))))
132103, 108oveq12d 7163 . . . . . . 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)⟩))
133132, 122fveq12d 6670 . . . . . 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)⟩))
134 df-ov 7148 . . . . . 6 ((1st𝑔)(⟨(1st𝑧), (2nd𝑧)⟩(2nd𝐸)⟨(1st𝑤), (2nd𝑤)⟩)(2nd𝑔)) = ((⟨(1st𝑧), (2nd𝑧)⟩(2nd𝐸)⟨(1st𝑤), (2nd𝑤)⟩)‘⟨(1st𝑔), (2nd𝑔)⟩)
135133, 134syl6eqr 2871 . . . . 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)))
136103fveq2d 6667 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑀𝑧) = (𝑀‘⟨(1st𝑧), (2nd𝑧)⟩))
137 df-ov 7148 . . . . . 6 ((1st𝑧)𝑀(2nd𝑧)) = (𝑀‘⟨(1st𝑧), (2nd𝑧)⟩)
138136, 137syl6eqr 2871 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑀𝑧) = ((1st𝑧)𝑀(2nd𝑧)))
139131, 135, 138oveq123d 7166 . . . 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))))
140101, 126, 1393eqtr4d 2863 . . 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)))
141140ralrimivvva 3189 . 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)))
142 eqid 2818 . . 3 ((𝑄 ×c 𝑂) Nat 𝑇) = ((𝑄 ×c 𝑂) Nat 𝑇)
143 eqid 2818 . . 3 (comp‘𝑇) = (comp‘𝑇)
144142, 33, 90, 29, 143, 37, 44isnat2 17206 . 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)))))
14571, 141, 144mpbir2and 709 1 (𝜑𝑀 ∈ (𝑍((𝑄 ×c 𝑂) Nat 𝑇)𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cun 3931  wss 3933  cop 4563   class class class wbr 5057  cmpt 5137   × cxp 5546  ran crn 5549  Rel wrel 5553   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  1st c1st 7676  2nd c2nd 7677  tpos ctpos 7880  Xcixp 8449  Basecbs 16471  Hom chom 16564  compcco 16565  Catccat 16923  Idccid 16924  Homf chomf 16925  oppCatcoppc 16969   Func cfunc 17112  func ccofu 17114   Nat cnat 17199   FuncCat cfuc 17200  SetCatcsetc 17323   ×c cxpc 17406   1stF c1stf 17407   2ndF c2ndf 17408   ⟨,⟩F cprf 17409   evalF cevlf 17447  HomFchof 17486  Yoncyon 17487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-tpos 7881  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12881  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-hom 16577  df-cco 16578  df-cat 16927  df-cid 16928  df-homf 16929  df-comf 16930  df-oppc 16970  df-ssc 17068  df-resc 17069  df-subc 17070  df-func 17116  df-cofu 17118  df-nat 17201  df-fuc 17202  df-setc 17324  df-xpc 17410  df-1stf 17411  df-2ndf 17412  df-prf 17413  df-evlf 17451  df-curf 17452  df-hof 17488  df-yon 17489
This theorem is referenced by:  yonedainv  17519
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