Theorem uppropd 49802

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 17935	. . 3 ⊢ (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
10	3	homfeqbas 17728	. . . 4 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
11	10	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 Func 𝐶)) → (Base‘𝐶) = (Base‘𝐷))
12	1	homfeqbas 17728	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
13	12	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) → (Base‘𝐴) = (Base‘𝐵))
14	13	adantr 484	. . . . . . 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 2762	. . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶)
16		eqid 2762	. . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
17		eqid 2762	. . . . . . . . 9 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
18	3	ad3antrrr 740	. . . . . . . . 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 782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) → 𝑤 ∈ (Base‘𝐶))
20	19	ad2antrr 736	. . . . . . . . 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 2762	. . . . . . . . . . . 12 ⊢ (Base‘𝐴) = (Base‘𝐴)
22		simprl 780	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) → 𝑓 ∈ (𝐴 Func 𝐶))
23	22	func1st2nd 49697	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) → (1st ‘𝑓)(𝐴 Func 𝐶)(2nd ‘𝑓))
24	21, 15, 23	funcf1 17899	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) → (1st ‘𝑓):(Base‘𝐴)⟶(Base‘𝐶))
25	24	adantr 484	. . . . . . . . . 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 7065	. . . . . . . . 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 17729	. . . . . . . 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 2762	. . . . . . . . . 10 ⊢ (Hom ‘𝐴) = (Hom ‘𝐴)
29		eqid 2762	. . . . . . . . . 10 ⊢ (Hom ‘𝐵) = (Hom ‘𝐵)
30	1	ad4antr 742	. . . . . . . . . 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 780	. . . . . . . . . . 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 736	. . . . . . . . . 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 778	. . . . . . . . . 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 17729	. . . . . . . . 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 2762	. . . . . . . . . . 11 ⊢ (comp‘𝐶) = (comp‘𝐶)
36		eqid 2762	. . . . . . . . . . 11 ⊢ (comp‘𝐷) = (comp‘𝐷)
37	18	ad2antrr 736	. . . . . . . . . . 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 744	. . . . . . . . . . 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 736	. . . . . . . . . . 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 7065	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ 𝑥 ∈ (Base‘𝐴)) → ((1st ‘𝑓)‘𝑥) ∈ (Base‘𝐶))
41	40	adantrr 727	. . . . . . . . . . . 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 740	. . . . . . . . . . 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 736	. . . . . . . . . . 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 782	. . . . . . . . . . . 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 740	. . . . . . . . . . 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 740	. . . . . . . . . . . . 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 17901	. . . . . . . . . . . 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 7065	. . . . . . . . . . 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 17740	. . . . . . . . . 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 2773	. . . . . . . . 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 49427	. . . . . . . 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 3326	. . . . . . 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 3326	. . . . . 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 587	. . . . 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 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ 𝑥 ∈ (Base‘𝐴)) → (Homf ‘𝐶) = (Homf ‘𝐷))
56		simplrr 787	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) ∧ 𝑥 ∈ (Base‘𝐴)) → 𝑤 ∈ (Base‘𝐶))
57	15, 16, 17, 55, 56, 40	homfeqval 17729	. . . . . . . . 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 2848	. . . . . . . 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 587	. . . . . . 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 2848	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑤 ∈ (Base‘𝐶))) → (𝑥 ∈ (Base‘𝐴) ↔ 𝑥 ∈ (Base‘𝐵)))
61	60	anbi1d 640	. . . . . . 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 281	. . . . . 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 640	. . . . 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 281	. . . 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 5166	. . 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 7470	. 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 49797	. 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 2762	. . 3 ⊢ (Base‘𝐵) = (Base‘𝐵)
69		eqid 2762	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
70	68, 69, 29, 17, 36	upfval 49797	. 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 2822	1 ⊢ (𝜑 → (𝐴 UP 𝐶) = (𝐵 UP 𝐷))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃!wreu 3365  ⟨cop 4588   class class class wbr 5100  {copab 5162  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396   ∈ cmpo 7398  1^st c1st 7968  2^nd c2nd 7969  Basecbs 17245  Hom chom 17297  compcco 17298  Hom_f chomf 17698  comp_fccomf 17699   Func cfunc 17887   UP cup 49794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-map 8810  df-ixp 8880  df-cat 17700  df-cid 17701  df-homf 17702  df-comf 17703  df-func 17891  df-up 49795

This theorem is referenced by:  lmdpropd  50278  cmdpropd  50279  cmddu  50289