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Theorem xpccatid 17905
Description: The product of two categories is a category. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
xpccat.t 𝑇 = (𝐶 ×c 𝐷)
xpccat.c (𝜑𝐶 ∈ Cat)
xpccat.d (𝜑𝐷 ∈ Cat)
xpccat.x 𝑋 = (Base‘𝐶)
xpccat.y 𝑌 = (Base‘𝐷)
xpccat.i 𝐼 = (Id‘𝐶)
xpccat.j 𝐽 = (Id‘𝐷)
Assertion
Ref Expression
xpccatid (𝜑 → (𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝐼𝑥), (𝐽𝑦)⟩)))
Distinct variable groups:   𝑥,𝑦,𝐼   𝑥,𝐽,𝑦   𝑥,𝐶,𝑦   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝐷,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝑇(𝑥,𝑦)

Proof of Theorem xpccatid
Dummy variables 𝑓 𝑔 𝑠 𝑡 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpccat.t . . . . 5 𝑇 = (𝐶 ×c 𝐷)
2 xpccat.x . . . . 5 𝑋 = (Base‘𝐶)
3 xpccat.y . . . . 5 𝑌 = (Base‘𝐷)
41, 2, 3xpcbas 17895 . . . 4 (𝑋 × 𝑌) = (Base‘𝑇)
54a1i 11 . . 3 (𝜑 → (𝑋 × 𝑌) = (Base‘𝑇))
6 eqidd 2739 . . 3 (𝜑 → (Hom ‘𝑇) = (Hom ‘𝑇))
7 eqidd 2739 . . 3 (𝜑 → (comp‘𝑇) = (comp‘𝑇))
81ovexi 7309 . . . 4 𝑇 ∈ V
98a1i 11 . . 3 (𝜑𝑇 ∈ V)
10 biid 260 . . 3 (((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣))) ↔ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣))))
11 eqid 2738 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
12 xpccat.i . . . . . 6 𝐼 = (Id‘𝐶)
13 xpccat.c . . . . . . 7 (𝜑𝐶 ∈ Cat)
1413adantr 481 . . . . . 6 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → 𝐶 ∈ Cat)
15 xp1st 7863 . . . . . . 7 (𝑡 ∈ (𝑋 × 𝑌) → (1st𝑡) ∈ 𝑋)
1615adantl 482 . . . . . 6 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (1st𝑡) ∈ 𝑋)
172, 11, 12, 14, 16catidcl 17391 . . . . 5 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (𝐼‘(1st𝑡)) ∈ ((1st𝑡)(Hom ‘𝐶)(1st𝑡)))
18 eqid 2738 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
19 xpccat.j . . . . . 6 𝐽 = (Id‘𝐷)
20 xpccat.d . . . . . . 7 (𝜑𝐷 ∈ Cat)
2120adantr 481 . . . . . 6 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → 𝐷 ∈ Cat)
22 xp2nd 7864 . . . . . . 7 (𝑡 ∈ (𝑋 × 𝑌) → (2nd𝑡) ∈ 𝑌)
2322adantl 482 . . . . . 6 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (2nd𝑡) ∈ 𝑌)
243, 18, 19, 21, 23catidcl 17391 . . . . 5 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (𝐽‘(2nd𝑡)) ∈ ((2nd𝑡)(Hom ‘𝐷)(2nd𝑡)))
2517, 24opelxpd 5627 . . . 4 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → ⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩ ∈ (((1st𝑡)(Hom ‘𝐶)(1st𝑡)) × ((2nd𝑡)(Hom ‘𝐷)(2nd𝑡))))
26 eqid 2738 . . . . 5 (Hom ‘𝑇) = (Hom ‘𝑇)
27 simpr 485 . . . . 5 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → 𝑡 ∈ (𝑋 × 𝑌))
281, 4, 11, 18, 26, 27, 27xpchom 17897 . . . 4 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (𝑡(Hom ‘𝑇)𝑡) = (((1st𝑡)(Hom ‘𝐶)(1st𝑡)) × ((2nd𝑡)(Hom ‘𝐷)(2nd𝑡))))
2925, 28eleqtrrd 2842 . . 3 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → ⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩ ∈ (𝑡(Hom ‘𝑇)𝑡))
30 fvex 6787 . . . . . . . 8 (𝐼‘(1st𝑡)) ∈ V
31 fvex 6787 . . . . . . . 8 (𝐽‘(2nd𝑡)) ∈ V
3230, 31op1st 7839 . . . . . . 7 (1st ‘⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩) = (𝐼‘(1st𝑡))
3332oveq1i 7285 . . . . . 6 ((1st ‘⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩)(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑡))(1st𝑓)) = ((𝐼‘(1st𝑡))(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑡))(1st𝑓))
3413adantr 481 . . . . . . 7 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝐶 ∈ Cat)
35 simpr1l 1229 . . . . . . . 8 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝑠 ∈ (𝑋 × 𝑌))
36 xp1st 7863 . . . . . . . 8 (𝑠 ∈ (𝑋 × 𝑌) → (1st𝑠) ∈ 𝑋)
3735, 36syl 17 . . . . . . 7 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (1st𝑠) ∈ 𝑋)
38 eqid 2738 . . . . . . 7 (comp‘𝐶) = (comp‘𝐶)
39 simpr1r 1230 . . . . . . . 8 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝑡 ∈ (𝑋 × 𝑌))
4039, 15syl 17 . . . . . . 7 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (1st𝑡) ∈ 𝑋)
41 simpr31 1262 . . . . . . . . 9 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡))
421, 4, 11, 18, 26, 35, 39xpchom 17897 . . . . . . . . 9 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (𝑠(Hom ‘𝑇)𝑡) = (((1st𝑠)(Hom ‘𝐶)(1st𝑡)) × ((2nd𝑠)(Hom ‘𝐷)(2nd𝑡))))
4341, 42eleqtrd 2841 . . . . . . . 8 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝑓 ∈ (((1st𝑠)(Hom ‘𝐶)(1st𝑡)) × ((2nd𝑠)(Hom ‘𝐷)(2nd𝑡))))
44 xp1st 7863 . . . . . . . 8 (𝑓 ∈ (((1st𝑠)(Hom ‘𝐶)(1st𝑡)) × ((2nd𝑠)(Hom ‘𝐷)(2nd𝑡))) → (1st𝑓) ∈ ((1st𝑠)(Hom ‘𝐶)(1st𝑡)))
4543, 44syl 17 . . . . . . 7 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (1st𝑓) ∈ ((1st𝑠)(Hom ‘𝐶)(1st𝑡)))
462, 11, 12, 34, 37, 38, 40, 45catlid 17392 . . . . . 6 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((𝐼‘(1st𝑡))(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑡))(1st𝑓)) = (1st𝑓))
4733, 46eqtrid 2790 . . . . 5 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((1st ‘⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩)(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑡))(1st𝑓)) = (1st𝑓))
4830, 31op2nd 7840 . . . . . . 7 (2nd ‘⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩) = (𝐽‘(2nd𝑡))
4948oveq1i 7285 . . . . . 6 ((2nd ‘⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩)(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑡))(2nd𝑓)) = ((𝐽‘(2nd𝑡))(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑡))(2nd𝑓))
5020adantr 481 . . . . . . 7 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝐷 ∈ Cat)
51 xp2nd 7864 . . . . . . . 8 (𝑠 ∈ (𝑋 × 𝑌) → (2nd𝑠) ∈ 𝑌)
5235, 51syl 17 . . . . . . 7 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (2nd𝑠) ∈ 𝑌)
53 eqid 2738 . . . . . . 7 (comp‘𝐷) = (comp‘𝐷)
5439, 22syl 17 . . . . . . 7 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (2nd𝑡) ∈ 𝑌)
55 xp2nd 7864 . . . . . . . 8 (𝑓 ∈ (((1st𝑠)(Hom ‘𝐶)(1st𝑡)) × ((2nd𝑠)(Hom ‘𝐷)(2nd𝑡))) → (2nd𝑓) ∈ ((2nd𝑠)(Hom ‘𝐷)(2nd𝑡)))
5643, 55syl 17 . . . . . . 7 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (2nd𝑓) ∈ ((2nd𝑠)(Hom ‘𝐷)(2nd𝑡)))
573, 18, 19, 50, 52, 53, 54, 56catlid 17392 . . . . . 6 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((𝐽‘(2nd𝑡))(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑡))(2nd𝑓)) = (2nd𝑓))
5849, 57eqtrid 2790 . . . . 5 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((2nd ‘⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩)(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑡))(2nd𝑓)) = (2nd𝑓))
5947, 58opeq12d 4812 . . . 4 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ⟨((1st ‘⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩)(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑡))(1st𝑓)), ((2nd ‘⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩)(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑡))(2nd𝑓))⟩ = ⟨(1st𝑓), (2nd𝑓)⟩)
60 eqid 2738 . . . . 5 (comp‘𝑇) = (comp‘𝑇)
6139, 29syldan 591 . . . . 5 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩ ∈ (𝑡(Hom ‘𝑇)𝑡))
621, 4, 26, 38, 53, 60, 35, 39, 39, 41, 61xpcco 17900 . . . 4 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑡)𝑓) = ⟨((1st ‘⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩)(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑡))(1st𝑓)), ((2nd ‘⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩)(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑡))(2nd𝑓))⟩)
63 1st2nd2 7870 . . . . 5 (𝑓 ∈ (((1st𝑠)(Hom ‘𝐶)(1st𝑡)) × ((2nd𝑠)(Hom ‘𝐷)(2nd𝑡))) → 𝑓 = ⟨(1st𝑓), (2nd𝑓)⟩)
6443, 63syl 17 . . . 4 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝑓 = ⟨(1st𝑓), (2nd𝑓)⟩)
6559, 62, 643eqtr4d 2788 . . 3 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑡)𝑓) = 𝑓)
6632oveq2i 7286 . . . . . 6 ((1st𝑔)(⟨(1st𝑡), (1st𝑡)⟩(comp‘𝐶)(1st𝑢))(1st ‘⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩)) = ((1st𝑔)(⟨(1st𝑡), (1st𝑡)⟩(comp‘𝐶)(1st𝑢))(𝐼‘(1st𝑡)))
67 simpr2l 1231 . . . . . . . 8 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝑢 ∈ (𝑋 × 𝑌))
68 xp1st 7863 . . . . . . . 8 (𝑢 ∈ (𝑋 × 𝑌) → (1st𝑢) ∈ 𝑋)
6967, 68syl 17 . . . . . . 7 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (1st𝑢) ∈ 𝑋)
70 simpr32 1263 . . . . . . . . 9 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢))
711, 4, 11, 18, 26, 39, 67xpchom 17897 . . . . . . . . 9 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (𝑡(Hom ‘𝑇)𝑢) = (((1st𝑡)(Hom ‘𝐶)(1st𝑢)) × ((2nd𝑡)(Hom ‘𝐷)(2nd𝑢))))
7270, 71eleqtrd 2841 . . . . . . . 8 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝑔 ∈ (((1st𝑡)(Hom ‘𝐶)(1st𝑢)) × ((2nd𝑡)(Hom ‘𝐷)(2nd𝑢))))
73 xp1st 7863 . . . . . . . 8 (𝑔 ∈ (((1st𝑡)(Hom ‘𝐶)(1st𝑢)) × ((2nd𝑡)(Hom ‘𝐷)(2nd𝑢))) → (1st𝑔) ∈ ((1st𝑡)(Hom ‘𝐶)(1st𝑢)))
7472, 73syl 17 . . . . . . 7 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (1st𝑔) ∈ ((1st𝑡)(Hom ‘𝐶)(1st𝑢)))
752, 11, 12, 34, 40, 38, 69, 74catrid 17393 . . . . . 6 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((1st𝑔)(⟨(1st𝑡), (1st𝑡)⟩(comp‘𝐶)(1st𝑢))(𝐼‘(1st𝑡))) = (1st𝑔))
7666, 75eqtrid 2790 . . . . 5 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((1st𝑔)(⟨(1st𝑡), (1st𝑡)⟩(comp‘𝐶)(1st𝑢))(1st ‘⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩)) = (1st𝑔))
7748oveq2i 7286 . . . . . 6 ((2nd𝑔)(⟨(2nd𝑡), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑢))(2nd ‘⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩)) = ((2nd𝑔)(⟨(2nd𝑡), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑢))(𝐽‘(2nd𝑡)))
78 xp2nd 7864 . . . . . . . 8 (𝑢 ∈ (𝑋 × 𝑌) → (2nd𝑢) ∈ 𝑌)
7967, 78syl 17 . . . . . . 7 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (2nd𝑢) ∈ 𝑌)
80 xp2nd 7864 . . . . . . . 8 (𝑔 ∈ (((1st𝑡)(Hom ‘𝐶)(1st𝑢)) × ((2nd𝑡)(Hom ‘𝐷)(2nd𝑢))) → (2nd𝑔) ∈ ((2nd𝑡)(Hom ‘𝐷)(2nd𝑢)))
8172, 80syl 17 . . . . . . 7 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (2nd𝑔) ∈ ((2nd𝑡)(Hom ‘𝐷)(2nd𝑢)))
823, 18, 19, 50, 54, 53, 79, 81catrid 17393 . . . . . 6 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((2nd𝑔)(⟨(2nd𝑡), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑢))(𝐽‘(2nd𝑡))) = (2nd𝑔))
8377, 82eqtrid 2790 . . . . 5 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((2nd𝑔)(⟨(2nd𝑡), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑢))(2nd ‘⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩)) = (2nd𝑔))
8476, 83opeq12d 4812 . . . 4 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ⟨((1st𝑔)(⟨(1st𝑡), (1st𝑡)⟩(comp‘𝐶)(1st𝑢))(1st ‘⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩)), ((2nd𝑔)(⟨(2nd𝑡), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑢))(2nd ‘⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩))⟩ = ⟨(1st𝑔), (2nd𝑔)⟩)
851, 4, 26, 38, 53, 60, 39, 39, 67, 61, 70xpcco 17900 . . . 4 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (𝑔(⟨𝑡, 𝑡⟩(comp‘𝑇)𝑢)⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩) = ⟨((1st𝑔)(⟨(1st𝑡), (1st𝑡)⟩(comp‘𝐶)(1st𝑢))(1st ‘⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩)), ((2nd𝑔)(⟨(2nd𝑡), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑢))(2nd ‘⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩))⟩)
86 1st2nd2 7870 . . . . 5 (𝑔 ∈ (((1st𝑡)(Hom ‘𝐶)(1st𝑢)) × ((2nd𝑡)(Hom ‘𝐷)(2nd𝑢))) → 𝑔 = ⟨(1st𝑔), (2nd𝑔)⟩)
8772, 86syl 17 . . . 4 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝑔 = ⟨(1st𝑔), (2nd𝑔)⟩)
8884, 85, 873eqtr4d 2788 . . 3 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (𝑔(⟨𝑡, 𝑡⟩(comp‘𝑇)𝑢)⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩) = 𝑔)
892, 11, 38, 34, 37, 40, 69, 45, 74catcocl 17394 . . . . 5 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((1st𝑔)(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑢))(1st𝑓)) ∈ ((1st𝑠)(Hom ‘𝐶)(1st𝑢)))
903, 18, 53, 50, 52, 54, 79, 56, 81catcocl 17394 . . . . 5 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((2nd𝑔)(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑢))(2nd𝑓)) ∈ ((2nd𝑠)(Hom ‘𝐷)(2nd𝑢)))
9189, 90opelxpd 5627 . . . 4 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ⟨((1st𝑔)(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑢))(1st𝑓)), ((2nd𝑔)(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑢))(2nd𝑓))⟩ ∈ (((1st𝑠)(Hom ‘𝐶)(1st𝑢)) × ((2nd𝑠)(Hom ‘𝐷)(2nd𝑢))))
921, 4, 26, 38, 53, 60, 35, 39, 67, 41, 70xpcco 17900 . . . 4 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓) = ⟨((1st𝑔)(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑢))(1st𝑓)), ((2nd𝑔)(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑢))(2nd𝑓))⟩)
931, 4, 11, 18, 26, 35, 67xpchom 17897 . . . 4 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (𝑠(Hom ‘𝑇)𝑢) = (((1st𝑠)(Hom ‘𝐶)(1st𝑢)) × ((2nd𝑠)(Hom ‘𝐷)(2nd𝑢))))
9491, 92, 933eltr4d 2854 . . 3 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓) ∈ (𝑠(Hom ‘𝑇)𝑢))
95 simpr2r 1232 . . . . . . . 8 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → 𝑣 ∈ (𝑋 × 𝑌))
96 xp1st 7863 . . . . . . . 8 (𝑣 ∈ (𝑋 × 𝑌) → (1st𝑣) ∈ 𝑋)
9795, 96syl 17 . . . . . . 7 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (1st𝑣) ∈ 𝑋)
98 simpr33 1264 . . . . . . . . 9 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ∈ (𝑢(Hom ‘𝑇)𝑣))
991, 4, 11, 18, 26, 67, 95xpchom 17897 . . . . . . . . 9 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (𝑢(Hom ‘𝑇)𝑣) = (((1st𝑢)(Hom ‘𝐶)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣))))
10098, 99eleqtrd 2841 . . . . . . . 8 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ∈ (((1st𝑢)(Hom ‘𝐶)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣))))
101 xp1st 7863 . . . . . . . 8 ( ∈ (((1st𝑢)(Hom ‘𝐶)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣))) → (1st) ∈ ((1st𝑢)(Hom ‘𝐶)(1st𝑣)))
102100, 101syl 17 . . . . . . 7 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (1st) ∈ ((1st𝑢)(Hom ‘𝐶)(1st𝑣)))
1032, 11, 38, 34, 37, 40, 69, 45, 74, 97, 102catass 17395 . . . . . 6 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (((1st)(⟨(1st𝑡), (1st𝑢)⟩(comp‘𝐶)(1st𝑣))(1st𝑔))(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑣))(1st𝑓)) = ((1st)(⟨(1st𝑠), (1st𝑢)⟩(comp‘𝐶)(1st𝑣))((1st𝑔)(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑢))(1st𝑓))))
1041, 4, 26, 38, 53, 60, 39, 67, 95, 70, 98xpcco 17900 . . . . . . . . 9 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔) = ⟨((1st)(⟨(1st𝑡), (1st𝑢)⟩(comp‘𝐶)(1st𝑣))(1st𝑔)), ((2nd)(⟨(2nd𝑡), (2nd𝑢)⟩(comp‘𝐷)(2nd𝑣))(2nd𝑔))⟩)
105104fveq2d 6778 . . . . . . . 8 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (1st ‘((⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔)) = (1st ‘⟨((1st)(⟨(1st𝑡), (1st𝑢)⟩(comp‘𝐶)(1st𝑣))(1st𝑔)), ((2nd)(⟨(2nd𝑡), (2nd𝑢)⟩(comp‘𝐷)(2nd𝑣))(2nd𝑔))⟩))
106 ovex 7308 . . . . . . . . 9 ((1st)(⟨(1st𝑡), (1st𝑢)⟩(comp‘𝐶)(1st𝑣))(1st𝑔)) ∈ V
107 ovex 7308 . . . . . . . . 9 ((2nd)(⟨(2nd𝑡), (2nd𝑢)⟩(comp‘𝐷)(2nd𝑣))(2nd𝑔)) ∈ V
108106, 107op1st 7839 . . . . . . . 8 (1st ‘⟨((1st)(⟨(1st𝑡), (1st𝑢)⟩(comp‘𝐶)(1st𝑣))(1st𝑔)), ((2nd)(⟨(2nd𝑡), (2nd𝑢)⟩(comp‘𝐷)(2nd𝑣))(2nd𝑔))⟩) = ((1st)(⟨(1st𝑡), (1st𝑢)⟩(comp‘𝐶)(1st𝑣))(1st𝑔))
109105, 108eqtrdi 2794 . . . . . . 7 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (1st ‘((⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔)) = ((1st)(⟨(1st𝑡), (1st𝑢)⟩(comp‘𝐶)(1st𝑣))(1st𝑔)))
110109oveq1d 7290 . . . . . 6 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((1st ‘((⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑣))(1st𝑓)) = (((1st)(⟨(1st𝑡), (1st𝑢)⟩(comp‘𝐶)(1st𝑣))(1st𝑔))(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑣))(1st𝑓)))
11192fveq2d 6778 . . . . . . . 8 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (1st ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓)) = (1st ‘⟨((1st𝑔)(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑢))(1st𝑓)), ((2nd𝑔)(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑢))(2nd𝑓))⟩))
112 ovex 7308 . . . . . . . . 9 ((1st𝑔)(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑢))(1st𝑓)) ∈ V
113 ovex 7308 . . . . . . . . 9 ((2nd𝑔)(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑢))(2nd𝑓)) ∈ V
114112, 113op1st 7839 . . . . . . . 8 (1st ‘⟨((1st𝑔)(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑢))(1st𝑓)), ((2nd𝑔)(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑢))(2nd𝑓))⟩) = ((1st𝑔)(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑢))(1st𝑓))
115111, 114eqtrdi 2794 . . . . . . 7 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (1st ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓)) = ((1st𝑔)(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑢))(1st𝑓)))
116115oveq2d 7291 . . . . . 6 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((1st)(⟨(1st𝑠), (1st𝑢)⟩(comp‘𝐶)(1st𝑣))(1st ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓))) = ((1st)(⟨(1st𝑠), (1st𝑢)⟩(comp‘𝐶)(1st𝑣))((1st𝑔)(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑢))(1st𝑓))))
117103, 110, 1163eqtr4d 2788 . . . . 5 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((1st ‘((⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑣))(1st𝑓)) = ((1st)(⟨(1st𝑠), (1st𝑢)⟩(comp‘𝐶)(1st𝑣))(1st ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓))))
118 xp2nd 7864 . . . . . . . 8 (𝑣 ∈ (𝑋 × 𝑌) → (2nd𝑣) ∈ 𝑌)
11995, 118syl 17 . . . . . . 7 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (2nd𝑣) ∈ 𝑌)
120 xp2nd 7864 . . . . . . . 8 ( ∈ (((1st𝑢)(Hom ‘𝐶)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣))) → (2nd) ∈ ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣)))
121100, 120syl 17 . . . . . . 7 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (2nd) ∈ ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣)))
1223, 18, 53, 50, 52, 54, 79, 56, 81, 119, 121catass 17395 . . . . . 6 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (((2nd)(⟨(2nd𝑡), (2nd𝑢)⟩(comp‘𝐷)(2nd𝑣))(2nd𝑔))(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑣))(2nd𝑓)) = ((2nd)(⟨(2nd𝑠), (2nd𝑢)⟩(comp‘𝐷)(2nd𝑣))((2nd𝑔)(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑢))(2nd𝑓))))
123104fveq2d 6778 . . . . . . . 8 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (2nd ‘((⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔)) = (2nd ‘⟨((1st)(⟨(1st𝑡), (1st𝑢)⟩(comp‘𝐶)(1st𝑣))(1st𝑔)), ((2nd)(⟨(2nd𝑡), (2nd𝑢)⟩(comp‘𝐷)(2nd𝑣))(2nd𝑔))⟩))
124106, 107op2nd 7840 . . . . . . . 8 (2nd ‘⟨((1st)(⟨(1st𝑡), (1st𝑢)⟩(comp‘𝐶)(1st𝑣))(1st𝑔)), ((2nd)(⟨(2nd𝑡), (2nd𝑢)⟩(comp‘𝐷)(2nd𝑣))(2nd𝑔))⟩) = ((2nd)(⟨(2nd𝑡), (2nd𝑢)⟩(comp‘𝐷)(2nd𝑣))(2nd𝑔))
125123, 124eqtrdi 2794 . . . . . . 7 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (2nd ‘((⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔)) = ((2nd)(⟨(2nd𝑡), (2nd𝑢)⟩(comp‘𝐷)(2nd𝑣))(2nd𝑔)))
126125oveq1d 7290 . . . . . 6 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((2nd ‘((⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑣))(2nd𝑓)) = (((2nd)(⟨(2nd𝑡), (2nd𝑢)⟩(comp‘𝐷)(2nd𝑣))(2nd𝑔))(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑣))(2nd𝑓)))
12792fveq2d 6778 . . . . . . . 8 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (2nd ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓)) = (2nd ‘⟨((1st𝑔)(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑢))(1st𝑓)), ((2nd𝑔)(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑢))(2nd𝑓))⟩))
128112, 113op2nd 7840 . . . . . . . 8 (2nd ‘⟨((1st𝑔)(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑢))(1st𝑓)), ((2nd𝑔)(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑢))(2nd𝑓))⟩) = ((2nd𝑔)(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑢))(2nd𝑓))
129127, 128eqtrdi 2794 . . . . . . 7 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (2nd ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓)) = ((2nd𝑔)(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑢))(2nd𝑓)))
130129oveq2d 7291 . . . . . 6 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((2nd)(⟨(2nd𝑠), (2nd𝑢)⟩(comp‘𝐷)(2nd𝑣))(2nd ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓))) = ((2nd)(⟨(2nd𝑠), (2nd𝑢)⟩(comp‘𝐷)(2nd𝑣))((2nd𝑔)(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑢))(2nd𝑓))))
131122, 126, 1303eqtr4d 2788 . . . . 5 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((2nd ‘((⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑣))(2nd𝑓)) = ((2nd)(⟨(2nd𝑠), (2nd𝑢)⟩(comp‘𝐷)(2nd𝑣))(2nd ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓))))
132117, 131opeq12d 4812 . . . 4 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ⟨((1st ‘((⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑣))(1st𝑓)), ((2nd ‘((⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑣))(2nd𝑓))⟩ = ⟨((1st)(⟨(1st𝑠), (1st𝑢)⟩(comp‘𝐶)(1st𝑣))(1st ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓))), ((2nd)(⟨(2nd𝑠), (2nd𝑢)⟩(comp‘𝐷)(2nd𝑣))(2nd ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓)))⟩)
1332, 11, 38, 34, 40, 69, 97, 74, 102catcocl 17394 . . . . . . 7 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((1st)(⟨(1st𝑡), (1st𝑢)⟩(comp‘𝐶)(1st𝑣))(1st𝑔)) ∈ ((1st𝑡)(Hom ‘𝐶)(1st𝑣)))
1343, 18, 53, 50, 54, 79, 119, 81, 121catcocl 17394 . . . . . . 7 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((2nd)(⟨(2nd𝑡), (2nd𝑢)⟩(comp‘𝐷)(2nd𝑣))(2nd𝑔)) ∈ ((2nd𝑡)(Hom ‘𝐷)(2nd𝑣)))
135133, 134opelxpd 5627 . . . . . 6 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ⟨((1st)(⟨(1st𝑡), (1st𝑢)⟩(comp‘𝐶)(1st𝑣))(1st𝑔)), ((2nd)(⟨(2nd𝑡), (2nd𝑢)⟩(comp‘𝐷)(2nd𝑣))(2nd𝑔))⟩ ∈ (((1st𝑡)(Hom ‘𝐶)(1st𝑣)) × ((2nd𝑡)(Hom ‘𝐷)(2nd𝑣))))
1361, 4, 11, 18, 26, 39, 95xpchom 17897 . . . . . 6 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (𝑡(Hom ‘𝑇)𝑣) = (((1st𝑡)(Hom ‘𝐶)(1st𝑣)) × ((2nd𝑡)(Hom ‘𝐷)(2nd𝑣))))
137135, 104, 1363eltr4d 2854 . . . . 5 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔) ∈ (𝑡(Hom ‘𝑇)𝑣))
1381, 4, 26, 38, 53, 60, 35, 39, 95, 41, 137xpcco 17900 . . . 4 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (((⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔)(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑣)𝑓) = ⟨((1st ‘((⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(1st𝑠), (1st𝑡)⟩(comp‘𝐶)(1st𝑣))(1st𝑓)), ((2nd ‘((⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔))(⟨(2nd𝑠), (2nd𝑡)⟩(comp‘𝐷)(2nd𝑣))(2nd𝑓))⟩)
1391, 4, 26, 38, 53, 60, 35, 67, 95, 94, 98xpcco 17900 . . . 4 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → ((⟨𝑠, 𝑢⟩(comp‘𝑇)𝑣)(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓)) = ⟨((1st)(⟨(1st𝑠), (1st𝑢)⟩(comp‘𝐶)(1st𝑣))(1st ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓))), ((2nd)(⟨(2nd𝑠), (2nd𝑢)⟩(comp‘𝐷)(2nd𝑣))(2nd ‘(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓)))⟩)
140132, 138, 1393eqtr4d 2788 . . 3 ((𝜑 ∧ ((𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌)) ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) ∧ (𝑓 ∈ (𝑠(Hom ‘𝑇)𝑡) ∧ 𝑔 ∈ (𝑡(Hom ‘𝑇)𝑢) ∧ ∈ (𝑢(Hom ‘𝑇)𝑣)))) → (((⟨𝑡, 𝑢⟩(comp‘𝑇)𝑣)𝑔)(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑣)𝑓) = ((⟨𝑠, 𝑢⟩(comp‘𝑇)𝑣)(𝑔(⟨𝑠, 𝑡⟩(comp‘𝑇)𝑢)𝑓)))
1415, 6, 7, 9, 10, 29, 65, 88, 94, 140iscatd2 17390 . 2 (𝜑 → (𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑡 ∈ (𝑋 × 𝑌) ↦ ⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩)))
142 vex 3436 . . . . . . . 8 𝑥 ∈ V
143 vex 3436 . . . . . . . 8 𝑦 ∈ V
144142, 143op1std 7841 . . . . . . 7 (𝑡 = ⟨𝑥, 𝑦⟩ → (1st𝑡) = 𝑥)
145144fveq2d 6778 . . . . . 6 (𝑡 = ⟨𝑥, 𝑦⟩ → (𝐼‘(1st𝑡)) = (𝐼𝑥))
146142, 143op2ndd 7842 . . . . . . 7 (𝑡 = ⟨𝑥, 𝑦⟩ → (2nd𝑡) = 𝑦)
147146fveq2d 6778 . . . . . 6 (𝑡 = ⟨𝑥, 𝑦⟩ → (𝐽‘(2nd𝑡)) = (𝐽𝑦))
148145, 147opeq12d 4812 . . . . 5 (𝑡 = ⟨𝑥, 𝑦⟩ → ⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩ = ⟨(𝐼𝑥), (𝐽𝑦)⟩)
149148mpompt 7388 . . . 4 (𝑡 ∈ (𝑋 × 𝑌) ↦ ⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝐼𝑥), (𝐽𝑦)⟩)
150149eqeq2i 2751 . . 3 ((Id‘𝑇) = (𝑡 ∈ (𝑋 × 𝑌) ↦ ⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩) ↔ (Id‘𝑇) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝐼𝑥), (𝐽𝑦)⟩))
151150anbi2i 623 . 2 ((𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑡 ∈ (𝑋 × 𝑌) ↦ ⟨(𝐼‘(1st𝑡)), (𝐽‘(2nd𝑡))⟩)) ↔ (𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝐼𝑥), (𝐽𝑦)⟩)))
152141, 151sylib 217 1 (𝜑 → (𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝐼𝑥), (𝐽𝑦)⟩)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  Vcvv 3432  cop 4567  cmpt 5157   × cxp 5587  cfv 6433  (class class class)co 7275  cmpo 7277  1st c1st 7829  2nd c2nd 7830  Basecbs 16912  Hom chom 16973  compcco 16974  Catccat 17373  Idccid 17374   ×c cxpc 17885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-fz 13240  df-struct 16848  df-slot 16883  df-ndx 16895  df-base 16913  df-hom 16986  df-cco 16987  df-cat 17377  df-cid 17378  df-xpc 17889
This theorem is referenced by:  xpcid  17906  xpccat  17907
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