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Theorem xpcpropd 18163
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 2732 . . 3 (𝐴 Γ—c 𝐢) = (𝐴 Γ—c 𝐢)
2 eqid 2732 . . 3 (Baseβ€˜π΄) = (Baseβ€˜π΄)
3 eqid 2732 . . 3 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
4 eqid 2732 . . 3 (Hom β€˜π΄) = (Hom β€˜π΄)
5 eqid 2732 . . 3 (Hom β€˜πΆ) = (Hom β€˜πΆ)
6 eqid 2732 . . 3 (compβ€˜π΄) = (compβ€˜π΄)
7 eqid 2732 . . 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 2733 . . 3 (πœ‘ β†’ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) = ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)))
111, 2, 3xpcbas 18132 . . . . 5 ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) = (Baseβ€˜(𝐴 Γ—c 𝐢))
12 eqid 2732 . . . . 5 (Hom β€˜(𝐴 Γ—c 𝐢)) = (Hom β€˜(𝐴 Γ—c 𝐢))
131, 11, 4, 5, 12xpchomfval 18133 . . . 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 2733 . . 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 18131 . 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 2732 . . 3 (𝐡 Γ—c 𝐷) = (𝐡 Γ—c 𝐷)
18 eqid 2732 . . 3 (Baseβ€˜π΅) = (Baseβ€˜π΅)
19 eqid 2732 . . 3 (Baseβ€˜π·) = (Baseβ€˜π·)
20 eqid 2732 . . 3 (Hom β€˜π΅) = (Hom β€˜π΅)
21 eqid 2732 . . 3 (Hom β€˜π·) = (Hom β€˜π·)
22 eqid 2732 . . 3 (compβ€˜π΅) = (compβ€˜π΅)
23 eqid 2732 . . 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 17642 . . . 4 (πœ‘ β†’ (Baseβ€˜π΄) = (Baseβ€˜π΅))
28 xpcpropd.3 . . . . 5 (πœ‘ β†’ (Homf β€˜πΆ) = (Homf β€˜π·))
2928homfeqbas 17642 . . . 4 (πœ‘ β†’ (Baseβ€˜πΆ) = (Baseβ€˜π·))
3027, 29xpeq12d 5707 . . 3 (πœ‘ β†’ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) = ((Baseβ€˜π΅) Γ— (Baseβ€˜π·)))
31263ad2ant1 1133 . . . . . . 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 8009 . . . . . . . 8 (𝑒 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) β†’ (1st β€˜π‘’) ∈ (Baseβ€˜π΄))
33323ad2ant2 1134 . . . . . . 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 8009 . . . . . . . 8 (𝑣 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) β†’ (1st β€˜π‘£) ∈ (Baseβ€˜π΄))
35343ad2ant3 1135 . . . . . . 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 17643 . . . . . 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 1133 . . . . . . 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 8010 . . . . . . . 8 (𝑒 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) β†’ (2nd β€˜π‘’) ∈ (Baseβ€˜πΆ))
39383ad2ant2 1134 . . . . . . 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 8010 . . . . . . . 8 (𝑣 ∈ ((Baseβ€˜π΄) Γ— (Baseβ€˜πΆ)) β†’ (2nd β€˜π‘£) ∈ (Baseβ€˜πΆ))
41403ad2ant3 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))
423, 5, 21, 37, 39, 41homfeqval 17643 . . . . . 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 5707 . . . . 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 7488 . . . 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 2784 . . 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 730 . . . . . . . . 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 730 . . . . . . . . 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 782 . . . . . . . . . . 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 8009 . . . . . . . . . . 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 8009 . . . . . . . . . 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 8010 . . . . . . . . . . 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 8009 . . . . . . . . . 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 774 . . . . . . . . . 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 8009 . . . . . . . . . 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 485 . . . . . . . . . . 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 8016 . . . . . . . . . . . . . . 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 6895 . . . . . . . . . . . . 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 7414 . . . . . . . . . . . . 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 2790 . . . . . . . . . . . 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 18134 . . . . . . . . . . . 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 2772 . . . . . . . . . . 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 2835 . . . . . . . . . 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 8009 . . . . . . . . . 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 767 . . . . . . . . . . 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 18134 . . . . . . . . . . 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 2835 . . . . . . . . . 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 8009 . . . . . . . . . 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 17654 . . . . . . . 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 730 . . . . . . . . 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 730 . . . . . . . . 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 8010 . . . . . . . . . 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 8010 . . . . . . . . . 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 8010 . . . . . . . . . 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 8010 . . . . . . . . . 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 8010 . . . . . . . . . 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 17654 . . . . . . . 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 4881 . . . . . . 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 1110 . . . . . 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 7488 . . . . 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 1110 . . . 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 7488 . . 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 18131 . 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 2775 1 (πœ‘ β†’ (𝐴 Γ—c 𝐢) = (𝐡 Γ—c 𝐷))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {ctp 4632  βŸ¨cop 4634   Γ— cxp 5674  β€˜cfv 6543  (class class class)co 7411   ∈ cmpo 7413  1st c1st 7975  2nd c2nd 7976  ndxcnx 17128  Basecbs 17146  Hom chom 17210  compcco 17211  Homf chomf 17612  compfccomf 17613   Γ—c cxpc 18122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-n0 12475  df-z 12561  df-dec 12680  df-uz 12825  df-fz 13487  df-struct 17082  df-slot 17117  df-ndx 17129  df-base 17147  df-hom 17223  df-cco 17224  df-homf 17616  df-comf 17617  df-xpc 18126
This theorem is referenced by:  curfpropd  18188  oppchofcl  18215
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