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Theorem xpcpropd 18102
Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same product category. (Contributed by Mario Carneiro, 17-Jan-2017.)
Hypotheses
Ref Expression
xpcpropd.1 (πœ‘ β†’ (Homf β€˜π΄) = (Homf β€˜π΅))
xpcpropd.2 (πœ‘ β†’ (compfβ€˜π΄) = (compfβ€˜π΅))
xpcpropd.3 (πœ‘ β†’ (Homf β€˜πΆ) = (Homf β€˜π·))
xpcpropd.4 (πœ‘ β†’ (compfβ€˜πΆ) = (compfβ€˜π·))
xpcpropd.a (πœ‘ β†’ 𝐴 ∈ 𝑉)
xpcpropd.b (πœ‘ β†’ 𝐡 ∈ 𝑉)
xpcpropd.c (πœ‘ β†’ 𝐢 ∈ 𝑉)
xpcpropd.d (πœ‘ β†’ 𝐷 ∈ 𝑉)
Assertion
Ref Expression
xpcpropd (πœ‘ β†’ (𝐴 Γ—c 𝐢) = (𝐡 Γ—c 𝐷))

Proof of Theorem xpcpropd
Dummy variables 𝑓 𝑔 𝑒 𝑣 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . 3 (𝐴 Γ—c 𝐢) = (𝐴 Γ—c 𝐢)
2 eqid 2733 . . 3 (Baseβ€˜π΄) = (Baseβ€˜π΄)
3 eqid 2733 . . 3 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
4 eqid 2733 . . 3 (Hom β€˜π΄) = (Hom β€˜π΄)
5 eqid 2733 . . 3 (Hom β€˜πΆ) = (Hom β€˜πΆ)
6 eqid 2733 . . 3 (compβ€˜π΄) = (compβ€˜π΄)
7 eqid 2733 . . 3 (compβ€˜πΆ) = (compβ€˜πΆ)
8 xpcpropd.a . . 3 (πœ‘ β†’ 𝐴 ∈ 𝑉)
9 xpcpropd.c . . 3 (πœ‘ β†’ 𝐢 ∈ 𝑉)
10 eqidd 2734 . . 3 (πœ‘ β†’ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) = ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))
111, 2, 3xpcbas 18071 . . . . 5 ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) = (Baseβ€˜(𝐴 Γ—c 𝐢))
12 eqid 2733 . . . . 5 (Hom β€˜(𝐴 Γ—c 𝐢)) = (Hom β€˜(𝐴 Γ—c 𝐢))
131, 11, 4, 5, 12xpchomfval 18072 . . . 4 (Hom β€˜(𝐴 Γ—c 𝐢)) = (𝑒 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)), 𝑣 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) ↦ (((1st β€˜π‘’)(Hom β€˜π΄)(1st β€˜π‘£)) Γ— ((2nd β€˜π‘’)(Hom β€˜πΆ)(2nd β€˜π‘£))))
1413a1i 11 . . 3 (πœ‘ β†’ (Hom β€˜(𝐴 Γ—c 𝐢)) = (𝑒 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)), 𝑣 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) ↦ (((1st β€˜π‘’)(Hom β€˜π΄)(1st β€˜π‘£)) Γ— ((2nd β€˜π‘’)(Hom β€˜πΆ)(2nd β€˜π‘£)))))
15 eqidd 2734 . . 3 (πœ‘ β†’ (π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))), 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) ↦ (𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦), 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜π΄)(1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜πΆ)(2nd β€˜π‘¦))(2nd β€˜π‘“))⟩)) = (π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))), 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) ↦ (𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦), 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜π΄)(1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜πΆ)(2nd β€˜π‘¦))(2nd β€˜π‘“))⟩)))
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15xpcval 18070 . 2 (πœ‘ β†’ (𝐴 Γ—c 𝐢) = {⟨(Baseβ€˜ndx), ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))⟩, ⟨(Hom β€˜ndx), (Hom β€˜(𝐴 Γ—c 𝐢))⟩, ⟨(compβ€˜ndx), (π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))), 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) ↦ (𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦), 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜π΄)(1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜πΆ)(2nd β€˜π‘¦))(2nd β€˜π‘“))⟩))⟩})
17 eqid 2733 . . 3 (𝐡 Γ—c 𝐷) = (𝐡 Γ—c 𝐷)
18 eqid 2733 . . 3 (Baseβ€˜π΅) = (Baseβ€˜π΅)
19 eqid 2733 . . 3 (Baseβ€˜π·) = (Baseβ€˜π·)
20 eqid 2733 . . 3 (Hom β€˜π΅) = (Hom β€˜π΅)
21 eqid 2733 . . 3 (Hom β€˜π·) = (Hom β€˜π·)
22 eqid 2733 . . 3 (compβ€˜π΅) = (compβ€˜π΅)
23 eqid 2733 . . 3 (compβ€˜π·) = (compβ€˜π·)
24 xpcpropd.b . . 3 (πœ‘ β†’ 𝐡 ∈ 𝑉)
25 xpcpropd.d . . 3 (πœ‘ β†’ 𝐷 ∈ 𝑉)
26 xpcpropd.1 . . . . 5 (πœ‘ β†’ (Homf β€˜π΄) = (Homf β€˜π΅))
2726homfeqbas 17581 . . . 4 (πœ‘ β†’ (Baseβ€˜π΄) = (Baseβ€˜π΅))
28 xpcpropd.3 . . . . 5 (πœ‘ β†’ (Homf β€˜πΆ) = (Homf β€˜π·))
2928homfeqbas 17581 . . . 4 (πœ‘ β†’ (Baseβ€˜πΆ) = (Baseβ€˜π·))
3027, 29xpeq12d 5665 . . 3 (πœ‘ β†’ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) = ((Baseβ€˜π΅) Γ— (Baseβ€˜π·)))
31263ad2ant1 1134 . . . . . . 7 ((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) ∧ 𝑣 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) β†’ (Homf β€˜π΄) = (Homf β€˜π΅))
32 xp1st 7954 . . . . . . . 8 (𝑒 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) β†’ (1st β€˜π‘’) ∈ (Baseβ€˜π΄))
33323ad2ant2 1135 . . . . . . 7 ((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) ∧ 𝑣 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) β†’ (1st β€˜π‘’) ∈ (Baseβ€˜π΄))
34 xp1st 7954 . . . . . . . 8 (𝑣 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) β†’ (1st β€˜π‘£) ∈ (Baseβ€˜π΄))
35343ad2ant3 1136 . . . . . . 7 ((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) ∧ 𝑣 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) β†’ (1st β€˜π‘£) ∈ (Baseβ€˜π΄))
362, 4, 20, 31, 33, 35homfeqval 17582 . . . . . 6 ((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) ∧ 𝑣 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) β†’ ((1st β€˜π‘’)(Hom β€˜π΄)(1st β€˜π‘£)) = ((1st β€˜π‘’)(Hom β€˜π΅)(1st β€˜π‘£)))
37283ad2ant1 1134 . . . . . . 7 ((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) ∧ 𝑣 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) β†’ (Homf β€˜πΆ) = (Homf β€˜π·))
38 xp2nd 7955 . . . . . . . 8 (𝑒 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) β†’ (2nd β€˜π‘’) ∈ (Baseβ€˜πΆ))
39383ad2ant2 1135 . . . . . . 7 ((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) ∧ 𝑣 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) β†’ (2nd β€˜π‘’) ∈ (Baseβ€˜πΆ))
40 xp2nd 7955 . . . . . . . 8 (𝑣 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) β†’ (2nd β€˜π‘£) ∈ (Baseβ€˜πΆ))
41403ad2ant3 1136 . . . . . . 7 ((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) ∧ 𝑣 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) β†’ (2nd β€˜π‘£) ∈ (Baseβ€˜πΆ))
423, 5, 21, 37, 39, 41homfeqval 17582 . . . . . 6 ((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) ∧ 𝑣 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) β†’ ((2nd β€˜π‘’)(Hom β€˜πΆ)(2nd β€˜π‘£)) = ((2nd β€˜π‘’)(Hom β€˜π·)(2nd β€˜π‘£)))
4336, 42xpeq12d 5665 . . . . 5 ((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) ∧ 𝑣 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) β†’ (((1st β€˜π‘’)(Hom β€˜π΄)(1st β€˜π‘£)) Γ— ((2nd β€˜π‘’)(Hom β€˜πΆ)(2nd β€˜π‘£))) = (((1st β€˜π‘’)(Hom β€˜π΅)(1st β€˜π‘£)) Γ— ((2nd β€˜π‘’)(Hom β€˜π·)(2nd β€˜π‘£))))
4443mpoeq3dva 7435 . . . 4 (πœ‘ β†’ (𝑒 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)), 𝑣 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) ↦ (((1st β€˜π‘’)(Hom β€˜π΄)(1st β€˜π‘£)) Γ— ((2nd β€˜π‘’)(Hom β€˜πΆ)(2nd β€˜π‘£)))) = (𝑒 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)), 𝑣 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) ↦ (((1st β€˜π‘’)(Hom β€˜π΅)(1st β€˜π‘£)) Γ— ((2nd β€˜π‘’)(Hom β€˜π·)(2nd β€˜π‘£)))))
4513, 44eqtrid 2785 . . 3 (πœ‘ β†’ (Hom β€˜(𝐴 Γ—c 𝐢)) = (𝑒 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)), 𝑣 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) ↦ (((1st β€˜π‘’)(Hom β€˜π΅)(1st β€˜π‘£)) Γ— ((2nd β€˜π‘’)(Hom β€˜π·)(2nd β€˜π‘£)))))
4626ad4antr 731 . . . . . . . . 9 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ (Homf β€˜π΄) = (Homf β€˜π΅))
47 xpcpropd.2 . . . . . . . . . 10 (πœ‘ β†’ (compfβ€˜π΄) = (compfβ€˜π΅))
4847ad4antr 731 . . . . . . . . 9 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ (compfβ€˜π΄) = (compfβ€˜π΅))
49 simp-4r 783 . . . . . . . . . . 11 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))))
50 xp1st 7954 . . . . . . . . . . 11 (π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) β†’ (1st β€˜π‘₯) ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))
5149, 50syl 17 . . . . . . . . . 10 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ (1st β€˜π‘₯) ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))
52 xp1st 7954 . . . . . . . . . 10 ((1st β€˜π‘₯) ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) β†’ (1st β€˜(1st β€˜π‘₯)) ∈ (Baseβ€˜π΄))
5351, 52syl 17 . . . . . . . . 9 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ (1st β€˜(1st β€˜π‘₯)) ∈ (Baseβ€˜π΄))
54 xp2nd 7955 . . . . . . . . . . 11 (π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) β†’ (2nd β€˜π‘₯) ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))
5549, 54syl 17 . . . . . . . . . 10 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ (2nd β€˜π‘₯) ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))
56 xp1st 7954 . . . . . . . . . 10 ((2nd β€˜π‘₯) ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) β†’ (1st β€˜(2nd β€˜π‘₯)) ∈ (Baseβ€˜π΄))
5755, 56syl 17 . . . . . . . . 9 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ (1st β€˜(2nd β€˜π‘₯)) ∈ (Baseβ€˜π΄))
58 simpllr 775 . . . . . . . . . 10 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))
59 xp1st 7954 . . . . . . . . . 10 (𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) β†’ (1st β€˜π‘¦) ∈ (Baseβ€˜π΄))
6058, 59syl 17 . . . . . . . . 9 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ (1st β€˜π‘¦) ∈ (Baseβ€˜π΄))
61 simpr 486 . . . . . . . . . . 11 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯))
62 1st2nd2 7961 . . . . . . . . . . . . . . 15 (π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) β†’ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩)
6349, 62syl 17 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩)
6463fveq2d 6847 . . . . . . . . . . . . 13 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯) = ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜βŸ¨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩))
65 df-ov 7361 . . . . . . . . . . . . 13 ((1st β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))(2nd β€˜π‘₯)) = ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜βŸ¨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩)
6664, 65eqtr4di 2791 . . . . . . . . . . . 12 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯) = ((1st β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))(2nd β€˜π‘₯)))
671, 11, 4, 5, 12, 51, 55xpchom 18073 . . . . . . . . . . . 12 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ ((1st β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))(2nd β€˜π‘₯)) = (((1st β€˜(1st β€˜π‘₯))(Hom β€˜π΄)(1st β€˜(2nd β€˜π‘₯))) Γ— ((2nd β€˜(1st β€˜π‘₯))(Hom β€˜πΆ)(2nd β€˜(2nd β€˜π‘₯)))))
6866, 67eqtrd 2773 . . . . . . . . . . 11 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯) = (((1st β€˜(1st β€˜π‘₯))(Hom β€˜π΄)(1st β€˜(2nd β€˜π‘₯))) Γ— ((2nd β€˜(1st β€˜π‘₯))(Hom β€˜πΆ)(2nd β€˜(2nd β€˜π‘₯)))))
6961, 68eleqtrd 2836 . . . . . . . . . 10 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ 𝑓 ∈ (((1st β€˜(1st β€˜π‘₯))(Hom β€˜π΄)(1st β€˜(2nd β€˜π‘₯))) Γ— ((2nd β€˜(1st β€˜π‘₯))(Hom β€˜πΆ)(2nd β€˜(2nd β€˜π‘₯)))))
70 xp1st 7954 . . . . . . . . . 10 (𝑓 ∈ (((1st β€˜(1st β€˜π‘₯))(Hom β€˜π΄)(1st β€˜(2nd β€˜π‘₯))) Γ— ((2nd β€˜(1st β€˜π‘₯))(Hom β€˜πΆ)(2nd β€˜(2nd β€˜π‘₯)))) β†’ (1st β€˜π‘“) ∈ ((1st β€˜(1st β€˜π‘₯))(Hom β€˜π΄)(1st β€˜(2nd β€˜π‘₯))))
7169, 70syl 17 . . . . . . . . 9 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ (1st β€˜π‘“) ∈ ((1st β€˜(1st β€˜π‘₯))(Hom β€˜π΄)(1st β€˜(2nd β€˜π‘₯))))
72 simplr 768 . . . . . . . . . . 11 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦))
731, 11, 4, 5, 12, 55, 58xpchom 18073 . . . . . . . . . . 11 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦) = (((1st β€˜(2nd β€˜π‘₯))(Hom β€˜π΄)(1st β€˜π‘¦)) Γ— ((2nd β€˜(2nd β€˜π‘₯))(Hom β€˜πΆ)(2nd β€˜π‘¦))))
7472, 73eleqtrd 2836 . . . . . . . . . 10 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ 𝑔 ∈ (((1st β€˜(2nd β€˜π‘₯))(Hom β€˜π΄)(1st β€˜π‘¦)) Γ— ((2nd β€˜(2nd β€˜π‘₯))(Hom β€˜πΆ)(2nd β€˜π‘¦))))
75 xp1st 7954 . . . . . . . . . 10 (𝑔 ∈ (((1st β€˜(2nd β€˜π‘₯))(Hom β€˜π΄)(1st β€˜π‘¦)) Γ— ((2nd β€˜(2nd β€˜π‘₯))(Hom β€˜πΆ)(2nd β€˜π‘¦))) β†’ (1st β€˜π‘”) ∈ ((1st β€˜(2nd β€˜π‘₯))(Hom β€˜π΄)(1st β€˜π‘¦)))
7674, 75syl 17 . . . . . . . . 9 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ (1st β€˜π‘”) ∈ ((1st β€˜(2nd β€˜π‘₯))(Hom β€˜π΄)(1st β€˜π‘¦)))
772, 4, 6, 22, 46, 48, 53, 57, 60, 71, 76comfeqval 17593 . . . . . . . 8 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ ((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜π΄)(1st β€˜π‘¦))(1st β€˜π‘“)) = ((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜π΅)(1st β€˜π‘¦))(1st β€˜π‘“)))
7828ad4antr 731 . . . . . . . . 9 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ (Homf β€˜πΆ) = (Homf β€˜π·))
79 xpcpropd.4 . . . . . . . . . 10 (πœ‘ β†’ (compfβ€˜πΆ) = (compfβ€˜π·))
8079ad4antr 731 . . . . . . . . 9 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ (compfβ€˜πΆ) = (compfβ€˜π·))
81 xp2nd 7955 . . . . . . . . . 10 ((1st β€˜π‘₯) ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) β†’ (2nd β€˜(1st β€˜π‘₯)) ∈ (Baseβ€˜πΆ))
8251, 81syl 17 . . . . . . . . 9 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ (2nd β€˜(1st β€˜π‘₯)) ∈ (Baseβ€˜πΆ))
83 xp2nd 7955 . . . . . . . . . 10 ((2nd β€˜π‘₯) ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) β†’ (2nd β€˜(2nd β€˜π‘₯)) ∈ (Baseβ€˜πΆ))
8455, 83syl 17 . . . . . . . . 9 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ (2nd β€˜(2nd β€˜π‘₯)) ∈ (Baseβ€˜πΆ))
85 xp2nd 7955 . . . . . . . . . 10 (𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) β†’ (2nd β€˜π‘¦) ∈ (Baseβ€˜πΆ))
8658, 85syl 17 . . . . . . . . 9 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ (2nd β€˜π‘¦) ∈ (Baseβ€˜πΆ))
87 xp2nd 7955 . . . . . . . . . 10 (𝑓 ∈ (((1st β€˜(1st β€˜π‘₯))(Hom β€˜π΄)(1st β€˜(2nd β€˜π‘₯))) Γ— ((2nd β€˜(1st β€˜π‘₯))(Hom β€˜πΆ)(2nd β€˜(2nd β€˜π‘₯)))) β†’ (2nd β€˜π‘“) ∈ ((2nd β€˜(1st β€˜π‘₯))(Hom β€˜πΆ)(2nd β€˜(2nd β€˜π‘₯))))
8869, 87syl 17 . . . . . . . . 9 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ (2nd β€˜π‘“) ∈ ((2nd β€˜(1st β€˜π‘₯))(Hom β€˜πΆ)(2nd β€˜(2nd β€˜π‘₯))))
89 xp2nd 7955 . . . . . . . . . 10 (𝑔 ∈ (((1st β€˜(2nd β€˜π‘₯))(Hom β€˜π΄)(1st β€˜π‘¦)) Γ— ((2nd β€˜(2nd β€˜π‘₯))(Hom β€˜πΆ)(2nd β€˜π‘¦))) β†’ (2nd β€˜π‘”) ∈ ((2nd β€˜(2nd β€˜π‘₯))(Hom β€˜πΆ)(2nd β€˜π‘¦)))
9074, 89syl 17 . . . . . . . . 9 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ (2nd β€˜π‘”) ∈ ((2nd β€˜(2nd β€˜π‘₯))(Hom β€˜πΆ)(2nd β€˜π‘¦)))
913, 5, 7, 23, 78, 80, 82, 84, 86, 88, 90comfeqval 17593 . . . . . . . 8 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜πΆ)(2nd β€˜π‘¦))(2nd β€˜π‘“)) = ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜π·)(2nd β€˜π‘¦))(2nd β€˜π‘“)))
9277, 91opeq12d 4839 . . . . . . 7 (((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦)) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜π΄)(1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜πΆ)(2nd β€˜π‘¦))(2nd β€˜π‘“))⟩ = ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜π΅)(1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜π·)(2nd β€˜π‘¦))(2nd β€˜π‘“))⟩)
93923impa 1111 . . . . . 6 ((((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦) ∧ 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯)) β†’ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜π΄)(1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜πΆ)(2nd β€˜π‘¦))(2nd β€˜π‘“))⟩ = ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜π΅)(1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜π·)(2nd β€˜π‘¦))(2nd β€˜π‘“))⟩)
9493mpoeq3dva 7435 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) β†’ (𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦), 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜π΄)(1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜πΆ)(2nd β€˜π‘¦))(2nd β€˜π‘“))⟩) = (𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦), 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜π΅)(1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜π·)(2nd β€˜π‘¦))(2nd β€˜π‘“))⟩))
95943impa 1111 . . . 4 ((πœ‘ ∧ π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) ∧ 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))) β†’ (𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦), 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜π΄)(1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜πΆ)(2nd β€˜π‘¦))(2nd β€˜π‘“))⟩) = (𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦), 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜π΅)(1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜π·)(2nd β€˜π‘¦))(2nd β€˜π‘“))⟩))
9695mpoeq3dva 7435 . . 3 (πœ‘ β†’ (π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))), 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) ↦ (𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦), 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜π΄)(1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜πΆ)(2nd β€˜π‘¦))(2nd β€˜π‘“))⟩)) = (π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))), 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) ↦ (𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦), 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜π΅)(1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜π·)(2nd β€˜π‘¦))(2nd β€˜π‘“))⟩)))
9717, 18, 19, 20, 21, 22, 23, 24, 25, 30, 45, 96xpcval 18070 . 2 (πœ‘ β†’ (𝐡 Γ—c 𝐷) = {⟨(Baseβ€˜ndx), ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))⟩, ⟨(Hom β€˜ndx), (Hom β€˜(𝐴 Γ—c 𝐢))⟩, ⟨(compβ€˜ndx), (π‘₯ ∈ (((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) Γ— ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ))), 𝑦 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) ↦ (𝑔 ∈ ((2nd β€˜π‘₯)(Hom β€˜(𝐴 Γ—c 𝐢))𝑦), 𝑓 ∈ ((Hom β€˜(𝐴 Γ—c 𝐢))β€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜π΄)(1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜πΆ)(2nd β€˜π‘¦))(2nd β€˜π‘“))⟩))⟩})
9816, 97eqtr4d 2776 1 (πœ‘ β†’ (𝐴 Γ—c 𝐢) = (𝐡 Γ—c 𝐷))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {ctp 4591  βŸ¨cop 4593   Γ— cxp 5632  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  1st c1st 7920  2nd c2nd 7921  ndxcnx 17070  Basecbs 17088  Hom chom 17149  compcco 17150  Homf chomf 17551  compfccomf 17552   Γ—c cxpc 18061
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-fz 13431  df-struct 17024  df-slot 17059  df-ndx 17071  df-base 17089  df-hom 17162  df-cco 17163  df-homf 17555  df-comf 17556  df-xpc 18065
This theorem is referenced by:  curfpropd  18127  oppchofcl  18154
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