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Theorem xpcval 16989
Description: Value of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.)
Hypotheses
Ref Expression
xpcval.t 𝑇 = (𝐶 ×c 𝐷)
xpcval.x 𝑋 = (Base‘𝐶)
xpcval.y 𝑌 = (Base‘𝐷)
xpcval.h 𝐻 = (Hom ‘𝐶)
xpcval.j 𝐽 = (Hom ‘𝐷)
xpcval.o1 · = (comp‘𝐶)
xpcval.o2 = (comp‘𝐷)
xpcval.c (𝜑𝐶𝑉)
xpcval.d (𝜑𝐷𝑊)
xpcval.b (𝜑𝐵 = (𝑋 × 𝑌))
xpcval.k (𝜑𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))))
xpcval.o (𝜑𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩)))
Assertion
Ref Expression
xpcval (𝜑𝑇 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩})
Distinct variable groups:   𝑓,𝑔,𝑢,𝑣,𝑥,𝑦,𝐵   𝜑,𝑓,𝑔,𝑢,𝑣,𝑥,𝑦   𝐶,𝑓,𝑔,𝑢,𝑣,𝑥,𝑦   𝐷,𝑓,𝑔,𝑢,𝑣,𝑥,𝑦   𝑓,𝐾,𝑔,𝑥,𝑦
Allowed substitution hints:   (𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)   𝑇(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)   · (𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)   𝐻(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)   𝐽(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)   𝐾(𝑣,𝑢)   𝑂(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)   𝑉(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)   𝑊(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)   𝑋(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)   𝑌(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)

Proof of Theorem xpcval
Dummy variables 𝑏 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpcval.t . 2 𝑇 = (𝐶 ×c 𝐷)
2 df-xpc 16984 . . . 4 ×c = (𝑟 ∈ V, 𝑠 ∈ V ↦ ((Base‘𝑟) × (Base‘𝑠)) / 𝑏(𝑢𝑏, 𝑣𝑏 ↦ (((1st𝑢)(Hom ‘𝑟)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣)))) / {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩))⟩})
32a1i 11 . . 3 (𝜑 → ×c = (𝑟 ∈ V, 𝑠 ∈ V ↦ ((Base‘𝑟) × (Base‘𝑠)) / 𝑏(𝑢𝑏, 𝑣𝑏 ↦ (((1st𝑢)(Hom ‘𝑟)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣)))) / {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩))⟩}))
4 fvex 6350 . . . . . 6 (Base‘𝑟) ∈ V
5 fvex 6350 . . . . . 6 (Base‘𝑠) ∈ V
64, 5xpex 7115 . . . . 5 ((Base‘𝑟) × (Base‘𝑠)) ∈ V
76a1i 11 . . . 4 ((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) → ((Base‘𝑟) × (Base‘𝑠)) ∈ V)
8 simprl 811 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) → 𝑟 = 𝐶)
98fveq2d 6344 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) → (Base‘𝑟) = (Base‘𝐶))
10 xpcval.x . . . . . . 7 𝑋 = (Base‘𝐶)
119, 10syl6eqr 2800 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) → (Base‘𝑟) = 𝑋)
12 simprr 813 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) → 𝑠 = 𝐷)
1312fveq2d 6344 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) → (Base‘𝑠) = (Base‘𝐷))
14 xpcval.y . . . . . . 7 𝑌 = (Base‘𝐷)
1513, 14syl6eqr 2800 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) → (Base‘𝑠) = 𝑌)
1611, 15xpeq12d 5285 . . . . 5 ((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) → ((Base‘𝑟) × (Base‘𝑠)) = (𝑋 × 𝑌))
17 xpcval.b . . . . . 6 (𝜑𝐵 = (𝑋 × 𝑌))
1817adantr 472 . . . . 5 ((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) → 𝐵 = (𝑋 × 𝑌))
1916, 18eqtr4d 2785 . . . 4 ((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) → ((Base‘𝑟) × (Base‘𝑠)) = 𝐵)
20 vex 3331 . . . . . . 7 𝑏 ∈ V
2120, 20mpt2ex 7403 . . . . . 6 (𝑢𝑏, 𝑣𝑏 ↦ (((1st𝑢)(Hom ‘𝑟)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣)))) ∈ V
2221a1i 11 . . . . 5 (((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (𝑢𝑏, 𝑣𝑏 ↦ (((1st𝑢)(Hom ‘𝑟)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣)))) ∈ V)
23 simpr 479 . . . . . . 7 (((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
24 simplrl 819 . . . . . . . . . . 11 (((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → 𝑟 = 𝐶)
2524fveq2d 6344 . . . . . . . . . 10 (((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (Hom ‘𝑟) = (Hom ‘𝐶))
26 xpcval.h . . . . . . . . . 10 𝐻 = (Hom ‘𝐶)
2725, 26syl6eqr 2800 . . . . . . . . 9 (((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (Hom ‘𝑟) = 𝐻)
2827oveqd 6818 . . . . . . . 8 (((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → ((1st𝑢)(Hom ‘𝑟)(1st𝑣)) = ((1st𝑢)𝐻(1st𝑣)))
29 simplrr 820 . . . . . . . . . . 11 (((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → 𝑠 = 𝐷)
3029fveq2d 6344 . . . . . . . . . 10 (((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (Hom ‘𝑠) = (Hom ‘𝐷))
31 xpcval.j . . . . . . . . . 10 𝐽 = (Hom ‘𝐷)
3230, 31syl6eqr 2800 . . . . . . . . 9 (((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (Hom ‘𝑠) = 𝐽)
3332oveqd 6818 . . . . . . . 8 (((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣)) = ((2nd𝑢)𝐽(2nd𝑣)))
3428, 33xpeq12d 5285 . . . . . . 7 (((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (((1st𝑢)(Hom ‘𝑟)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣))) = (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))
3523, 23, 34mpt2eq123dv 6870 . . . . . 6 (((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (𝑢𝑏, 𝑣𝑏 ↦ (((1st𝑢)(Hom ‘𝑟)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣)))) = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))))
36 xpcval.k . . . . . . 7 (𝜑𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))))
3736ad2antrr 764 . . . . . 6 (((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → 𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))))
3835, 37eqtr4d 2785 . . . . 5 (((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (𝑢𝑏, 𝑣𝑏 ↦ (((1st𝑢)(Hom ‘𝑟)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣)))) = 𝐾)
39 simplr 809 . . . . . . 7 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → 𝑏 = 𝐵)
4039opeq2d 4548 . . . . . 6 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
41 simpr 479 . . . . . . 7 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → = 𝐾)
4241opeq2d 4548 . . . . . 6 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → ⟨(Hom ‘ndx), ⟩ = ⟨(Hom ‘ndx), 𝐾⟩)
4339, 39xpeq12d 5285 . . . . . . . . 9 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
4441oveqd 6818 . . . . . . . . . 10 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → ((2nd𝑥)𝑦) = ((2nd𝑥)𝐾𝑦))
4541fveq1d 6342 . . . . . . . . . 10 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → (𝑥) = (𝐾𝑥))
4624adantr 472 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → 𝑟 = 𝐶)
4746fveq2d 6344 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → (comp‘𝑟) = (comp‘𝐶))
48 xpcval.o1 . . . . . . . . . . . . . 14 · = (comp‘𝐶)
4947, 48syl6eqr 2800 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → (comp‘𝑟) = · )
5049oveqd 6818 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → (⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦)) = (⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦)))
5150oveqd 6818 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)) = ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)))
5229adantr 472 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → 𝑠 = 𝐷)
5352fveq2d 6344 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → (comp‘𝑠) = (comp‘𝐷))
54 xpcval.o2 . . . . . . . . . . . . . 14 = (comp‘𝐷)
5553, 54syl6eqr 2800 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → (comp‘𝑠) = )
5655oveqd 6818 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → (⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦)) = (⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦)))
5756oveqd 6818 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓)) = ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓)))
5851, 57opeq12d 4549 . . . . . . . . . 10 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩ = ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩)
5944, 45, 58mpt2eq123dv 6870 . . . . . . . . 9 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩) = (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩))
6043, 39, 59mpt2eq123dv 6870 . . . . . . . 8 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩)) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩)))
61 xpcval.o . . . . . . . . 9 (𝜑𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩)))
6261ad3antrrr 768 . . . . . . . 8 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩)))
6360, 62eqtr4d 2785 . . . . . . 7 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩)) = 𝑂)
6463opeq2d 4548 . . . . . 6 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩))⟩ = ⟨(comp‘ndx), 𝑂⟩)
6540, 42, 64tpeq123d 4415 . . . . 5 ((((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ = 𝐾) → {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩})
6622, 38, 65csbied2 3690 . . . 4 (((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (𝑢𝑏, 𝑣𝑏 ↦ (((1st𝑢)(Hom ‘𝑟)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣)))) / {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩})
677, 19, 66csbied2 3690 . . 3 ((𝜑 ∧ (𝑟 = 𝐶𝑠 = 𝐷)) → ((Base‘𝑟) × (Base‘𝑠)) / 𝑏(𝑢𝑏, 𝑣𝑏 ↦ (((1st𝑢)(Hom ‘𝑟)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣)))) / {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩})
68 xpcval.c . . . 4 (𝜑𝐶𝑉)
69 elex 3340 . . . 4 (𝐶𝑉𝐶 ∈ V)
7068, 69syl 17 . . 3 (𝜑𝐶 ∈ V)
71 xpcval.d . . . 4 (𝜑𝐷𝑊)
72 elex 3340 . . . 4 (𝐷𝑊𝐷 ∈ V)
7371, 72syl 17 . . 3 (𝜑𝐷 ∈ V)
74 tpex 7110 . . . 4 {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩} ∈ V
7574a1i 11 . . 3 (𝜑 → {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩} ∈ V)
763, 67, 70, 73, 75ovmpt2d 6941 . 2 (𝜑 → (𝐶 ×c 𝐷) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩})
771, 76syl5eq 2794 1 (𝜑𝑇 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1620  wcel 2127  Vcvv 3328  csb 3662  {ctp 4313  cop 4315   × cxp 5252  cfv 6037  (class class class)co 6801  cmpt2 6803  1st c1st 7319  2nd c2nd 7320  ndxcnx 16027  Basecbs 16030  Hom chom 16125  compcco 16126   ×c cxpc 16980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-1st 7321  df-2nd 7322  df-xpc 16984
This theorem is referenced by:  xpcbas  16990  xpchomfval  16991  xpccofval  16994  catcxpccl  17019  xpcpropd  17020
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