Theorem uppropd 50433

Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same universal pairs. (Contributed by Zhi Wang, 20-Nov-2025.)

Hypotheses

Ref	Expression
uppropd.1	⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
uppropd.2	⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))
uppropd.3	⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
uppropd.4	⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
uppropd.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
uppropd.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)
uppropd.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
uppropd.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)

Assertion

Ref	Expression
uppropd	⊢ (𝜑 → (𝐴 UP 𝐶) = (𝐵 UP 𝐷))

Proof of Theorem uppropd

Dummy variables 𝑓 𝑔 𝑘 𝑚 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uppropd.1	. . . 4 ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
2		uppropd.2	. . . 4 ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))
3		uppropd.3	. . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
4		uppropd.4	. . . 4 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
5		uppropd.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		uppropd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
7		uppropd.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
8		uppropd.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
9	1, 2, 3, 4, 5, 6, 7, 8	funcpropd 18086	. . 3 ⊢ (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
10	3	homfeqbas 17879	. . . 4 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
11	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 Func 𝐶)) → (Base‘𝐶) = (Base‘𝐷))
12	1	homfeqbas 17879	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
13	12	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) → (Base‘𝐴) = (Base‘𝐵))
14	13	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) → (Base‘𝐴) = (Base‘𝐵))
15		eqid 2760	. . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶)
16		eqid 2760	. . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
17		eqid 2760	. . . . . . . . 9 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
18	3	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) ∧ 𝑦 ∈ (Base‘𝐴)) → (Homf ‘𝐶) = (Homf ‘𝐷))
19		simprr 785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) → 𝑤 ∈ (Base‘𝐶))
20	19	ad2antrr 739	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑤 ∈ (Base‘𝐶))
21		eqid 2760	. . . . . . . . . . . 12 ⊢ (Base‘𝐴) = (Base‘𝐴)
22		simprl 783	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) → 𝑓 ∈ (𝐴 Func 𝐶))
23	22	func1st2nd 50326	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) → (1st ‘𝑓)(𝐴 Func 𝐶)(2nd ‘𝑓))
24	21, 15, 23	funcf1 18050	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) → (1st ‘𝑓):(Base‘𝐴)⟶(Base‘𝐶))
25	24	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) → (1st ‘𝑓):(Base‘𝐴)⟶(Base‘𝐶))
26	25	ffvelcdmda 7089	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) ∧ 𝑦 ∈ (Base‘𝐴)) → ((1st ‘𝑓)‘𝑦) ∈ (Base‘𝐶))
27	15, 16, 17, 18, 20, 26	homfeqval 17880	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦)) = (𝑤(Hom ‘𝐷)((1st ‘𝑓)‘𝑦)))
28		eqid 2760	. . . . . . . . . 10 ⊢ (Hom ‘𝐴) = (Hom ‘𝐴)
29		eqid 2760	. . . . . . . . . 10 ⊢ (Hom ‘𝐵) = (Hom ‘𝐵)
30	1	ad4antr 745	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ 𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))) → (Homf ‘𝐴) = (Homf ‘𝐵))
31		simprl 783	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) → 𝑥 ∈ (Base‘𝐴))
32	31	ad2antrr 739	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ 𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))) → 𝑥 ∈ (Base‘𝐴))
33		simplr 781	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ 𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))) → 𝑦 ∈ (Base‘𝐴))
34	21, 28, 29, 30, 32, 33	homfeqval 17880	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ 𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))) → (𝑥(Hom ‘𝐴)𝑦) = (𝑥(Hom ‘𝐵)𝑦))
35		eqid 2760	. . . . . . . . . . 11 ⊢ (comp‘𝐶) = (comp‘𝐶)
36		eqid 2760	. . . . . . . . . . 11 ⊢ (comp‘𝐷) = (comp‘𝐷)
37	18	ad2antrr 739	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ 𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))) ∧ 𝑘 ∈ (𝑥(Hom ‘𝐴)𝑦)) → (Homf ‘𝐶) = (Homf ‘𝐷))
38	4	ad5antr 747	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ 𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))) ∧ 𝑘 ∈ (𝑥(Hom ‘𝐴)𝑦)) → (compf‘𝐶) = (compf‘𝐷))
39	20	ad2antrr 739	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ 𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))) ∧ 𝑘 ∈ (𝑥(Hom ‘𝐴)𝑦)) → 𝑤 ∈ (Base‘𝐶))
40	24	ffvelcdmda 7089	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ 𝑥 ∈ (Base‘𝐴)) → ((1st ‘𝑓)‘𝑥) ∈ (Base‘𝐶))
41	40	adantrr 730	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) → ((1st ‘𝑓)‘𝑥) ∈ (Base‘𝐶))
42	41	ad3antrrr 743	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ 𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))) ∧ 𝑘 ∈ (𝑥(Hom ‘𝐴)𝑦)) → ((1st ‘𝑓)‘𝑥) ∈ (Base‘𝐶))
43	26	ad2antrr 739	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ 𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))) ∧ 𝑘 ∈ (𝑥(Hom ‘𝐴)𝑦)) → ((1st ‘𝑓)‘𝑦) ∈ (Base‘𝐶))
44		simprr 785	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) → 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))
45	44	ad3antrrr 743	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ 𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))) ∧ 𝑘 ∈ (𝑥(Hom ‘𝐴)𝑦)) → 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))
46	23	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ 𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))) → (1st ‘𝑓)(𝐴 Func 𝐶)(2nd ‘𝑓))
47	21, 28, 16, 46, 32, 33	funcf2 18052	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ 𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))) → (𝑥(2nd ‘𝑓)𝑦):(𝑥(Hom ‘𝐴)𝑦)⟶(((1st ‘𝑓)‘𝑥)(Hom ‘𝐶)((1st ‘𝑓)‘𝑦)))
48	47	ffvelcdmda 7089	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ 𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))) ∧ 𝑘 ∈ (𝑥(Hom ‘𝐴)𝑦)) → ((𝑥(2nd ‘𝑓)𝑦)‘𝑘) ∈ (((1st ‘𝑓)‘𝑥)(Hom ‘𝐶)((1st ‘𝑓)‘𝑦)))
49	15, 16, 35, 36, 37, 38, 39, 42, 43, 45, 48	comfeqval 17891	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ 𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))) ∧ 𝑘 ∈ (𝑥(Hom ‘𝐴)𝑦)) → (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐶)((1st ‘𝑓)‘𝑦))𝑚) = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐷)((1st ‘𝑓)‘𝑦))𝑚))
50	49	eqeq2d 2771	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ 𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))) ∧ 𝑘 ∈ (𝑥(Hom ‘𝐴)𝑦)) → (𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐶)((1st ‘𝑓)‘𝑦))𝑚) ↔ 𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐷)((1st ‘𝑓)‘𝑦))𝑚)))
51	34, 50	reueqbidva 50052	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ 𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))) → (∃!𝑘 ∈ (𝑥(Hom ‘𝐴)𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐶)((1st ‘𝑓)‘𝑦))𝑚) ↔ ∃!𝑘 ∈ (𝑥(Hom ‘𝐵)𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐷)((1st ‘𝑓)‘𝑦))𝑚)))
52	27, 51	raleqbidva 3325	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) ∧ 𝑦 ∈ (Base‘𝐴)) → (∀𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥(Hom ‘𝐴)𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐶)((1st ‘𝑓)‘𝑦))𝑚) ↔ ∀𝑔 ∈ (𝑤(Hom ‘𝐷)((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥(Hom ‘𝐵)𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐷)((1st ‘𝑓)‘𝑦))𝑚)))
53	14, 52	raleqbidva 3325	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)))) → (∀𝑦 ∈ (Base‘𝐴)∀𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥(Hom ‘𝐴)𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐶)((1st ‘𝑓)‘𝑦))𝑚) ↔ ∀𝑦 ∈ (Base‘𝐵)∀𝑔 ∈ (𝑤(Hom ‘𝐷)((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥(Hom ‘𝐵)𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐷)((1st ‘𝑓)‘𝑦))𝑚)))
54	53	pm5.32da 590	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) → (((𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ (Base‘𝐴)∀𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥(Hom ‘𝐴)𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐶)((1st ‘𝑓)‘𝑦))𝑚)) ↔ ((𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ (Base‘𝐵)∀𝑔 ∈ (𝑤(Hom ‘𝐷)((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥(Hom ‘𝐵)𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐷)((1st ‘𝑓)‘𝑦))𝑚))))
55	3	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ 𝑥 ∈ (Base‘𝐴)) → (Homf ‘𝐶) = (Homf ‘𝐷))
56		simplrr 790	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ 𝑥 ∈ (Base‘𝐴)) → 𝑤 ∈ (Base‘𝐶))
57	15, 16, 17, 55, 56, 40	homfeqval 17880	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)) = (𝑤(Hom ‘𝐷)((1st ‘𝑓)‘𝑥)))
58	57	eleq2d 2846	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥)) ↔ 𝑚 ∈ (𝑤(Hom ‘𝐷)((1st ‘𝑓)‘𝑥))))
59	58	pm5.32da 590	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) → ((𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥))) ↔ (𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐷)((1st ‘𝑓)‘𝑥)))))
60	13	eleq2d 2846	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) → (𝑥 ∈ (Base‘𝐴) ↔ 𝑥 ∈ (Base‘𝐵)))
61	60	anbi1d 643	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) → ((𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐷)((1st ‘𝑓)‘𝑥))) ↔ (𝑥 ∈ (Base‘𝐵) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐷)((1st ‘𝑓)‘𝑥)))))
62	59, 61	bitrd 282	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) → ((𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥))) ↔ (𝑥 ∈ (Base‘𝐵) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐷)((1st ‘𝑓)‘𝑥)))))
63	62	anbi1d 643	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) → (((𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ (Base‘𝐵)∀𝑔 ∈ (𝑤(Hom ‘𝐷)((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥(Hom ‘𝐵)𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐷)((1st ‘𝑓)‘𝑦))𝑚)) ↔ ((𝑥 ∈ (Base‘𝐵) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐷)((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ (Base‘𝐵)∀𝑔 ∈ (𝑤(Hom ‘𝐷)((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥(Hom ‘𝐵)𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐷)((1st ‘𝑓)‘𝑦))𝑚))))
64	54, 63	bitrd 282	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) → (((𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ (Base‘𝐴)∀𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥(Hom ‘𝐴)𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐶)((1st ‘𝑓)‘𝑦))𝑚)) ↔ ((𝑥 ∈ (Base‘𝐵) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐷)((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ (Base‘𝐵)∀𝑔 ∈ (𝑤(Hom ‘𝐷)((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥(Hom ‘𝐵)𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐷)((1st ‘𝑓)‘𝑦))𝑚))))
65	64	opabbidv 5170	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) → {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ (Base‘𝐴)∀𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥(Hom ‘𝐴)𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐶)((1st ‘𝑓)‘𝑦))𝑚))} = {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ (Base‘𝐵) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐷)((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ (Base‘𝐵)∀𝑔 ∈ (𝑤(Hom ‘𝐷)((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥(Hom ‘𝐵)𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐷)((1st ‘𝑓)‘𝑦))𝑚))})
66	9, 11, 65	mpoeq123dva 7498	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐴 Func 𝐶), 𝑤 ∈ (Base‘𝐶) ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ (Base‘𝐴)∀𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥(Hom ‘𝐴)𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐶)((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐵 Func 𝐷), 𝑤 ∈ (Base‘𝐷) ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ (Base‘𝐵) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐷)((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ (Base‘𝐵)∀𝑔 ∈ (𝑤(Hom ‘𝐷)((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥(Hom ‘𝐵)𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐷)((1st ‘𝑓)‘𝑦))𝑚))}))
67	21, 15, 28, 16, 35	upfval 50428	. 2 ⊢ (𝐴 UP 𝐶) = (𝑓 ∈ (𝐴 Func 𝐶), 𝑤 ∈ (Base‘𝐶) ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ (Base‘𝐴) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ (Base‘𝐴)∀𝑔 ∈ (𝑤(Hom ‘𝐶)((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥(Hom ‘𝐴)𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐶)((1st ‘𝑓)‘𝑦))𝑚))})
68		eqid 2760	. . 3 ⊢ (Base‘𝐵) = (Base‘𝐵)
69		eqid 2760	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
70	68, 69, 29, 17, 36	upfval 50428	. 2 ⊢ (𝐵 UP 𝐷) = (𝑓 ∈ (𝐵 Func 𝐷), 𝑤 ∈ (Base‘𝐷) ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ (Base‘𝐵) ∧ 𝑚 ∈ (𝑤(Hom ‘𝐷)((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ (Base‘𝐵)∀𝑔 ∈ (𝑤(Hom ‘𝐷)((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥(Hom ‘𝐵)𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩(comp‘𝐷)((1st ‘𝑓)‘𝑦))𝑚))})
71	66, 67, 70	3eqtr4g 2820	1 ⊢ (𝜑 → (𝐴 UP 𝐶) = (𝐵 UP 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃!wreu 3363  ⟨cop 4834   class class class wbr 5102  {copab 5166  ⟶wf 6539  ‘cfv 6543  (class class class)co 7424   ∈ cmpo 7426  1^st c1st 8003  2^nd c2nd 8004  Basecbs 17396  Hom chom 17448  compcco 17449  Hom_f chomf 17849  comp_fccomf 17850   Func cfunc 18038   UP cup 50425

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-map 8857  df-ixp 8934  df-cat 17851  df-cid 17852  df-homf 17853  df-comf 17854  df-func 18042  df-up 50426

This theorem is used by:  lmdpropd  50976  cmdpropd  50977  cmddu  50987