Theorem upeu2 49647

Description: Generate new universal morphism through isomorphism from existing universal object. (Contributed by Zhi Wang, 20-Sep-2025.)

Hypotheses

Ref	Expression
upcic.b	⊢ 𝐵 = (Base‘𝐷)
upcic.c	⊢ 𝐶 = (Base‘𝐸)
upcic.h	⊢ 𝐻 = (Hom ‘𝐷)
upcic.j	⊢ 𝐽 = (Hom ‘𝐸)
upcic.o	⊢ 𝑂 = (comp‘𝐸)
upcic.f	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
upcic.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
upcic.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
upcic.z	⊢ (𝜑 → 𝑍 ∈ 𝐶)
upcic.m	⊢ (𝜑 → 𝑀 ∈ (𝑍𝐽(𝐹‘𝑋)))
upcic.1	⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ∀𝑓 ∈ (𝑍𝐽(𝐹‘𝑤))∃!𝑘 ∈ (𝑋𝐻𝑤)𝑓 = (((𝑋𝐺𝑤)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑤))𝑀))
upeu2.i	⊢ 𝐼 = (Iso‘𝐷)
upeu2.k	⊢ (𝜑 → 𝐾 ∈ (𝑋𝐼𝑌))
upeu2.n	⊢ (𝜑 → 𝑁 = (((𝑋𝐺𝑌)‘𝐾)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))

Assertion

Ref	Expression
upeu2	⊢ (𝜑 → (𝑁 ∈ (𝑍𝐽(𝐹‘𝑌)) ∧ ∀𝑣 ∈ 𝐵 ∀𝑔 ∈ (𝑍𝐽(𝐹‘𝑣))∃!𝑙 ∈ (𝑌𝐻𝑣)𝑔 = (((𝑌𝐺𝑣)‘𝑙)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑣))𝑁)))

Distinct variable groups:   𝐵,𝑔,𝑙   𝑤,𝐵   𝐷,𝑙   𝑓,𝐹,𝑘,𝑤   𝐹,𝑙   𝑓,𝐺,𝑘,𝑤   𝐺,𝑙   𝑓,𝐻,𝑘,𝑤   𝐻,𝑙   𝑓,𝐽,𝑤   𝐽,𝑙   𝐾,𝑙   𝑓,𝑀,𝑘,𝑤   𝑀,𝑙   𝑓,𝑂,𝑘,𝑤   𝑂,𝑙   𝑓,𝑋,𝑘,𝑤   𝑋,𝑙   𝑌,𝑙   𝑓,𝑍,𝑘,𝑤   𝑍,𝑙   𝑓,𝑔,𝑣,𝑘   𝜑,𝑔,𝑙,𝑣   𝑤,𝑣

Allowed substitution hints:   𝜑(𝑤,𝑓,𝑘)   𝐵(𝑣,𝑓,𝑘)   𝐶(𝑤,𝑣,𝑓,𝑔,𝑘,𝑙)   𝐷(𝑤,𝑣,𝑓,𝑔,𝑘)   𝐸(𝑤,𝑣,𝑓,𝑔,𝑘,𝑙)   𝐹(𝑣,𝑔)   𝐺(𝑣,𝑔)   𝐻(𝑣,𝑔)   𝐼(𝑤,𝑣,𝑓,𝑔,𝑘,𝑙)   𝐽(𝑣,𝑔,𝑘)   𝐾(𝑤,𝑣,𝑓,𝑔,𝑘)   𝑀(𝑣,𝑔)   𝑁(𝑤,𝑣,𝑓,𝑔,𝑘,𝑙)   𝑂(𝑣,𝑔)   𝑋(𝑣,𝑔)   𝑌(𝑤,𝑣,𝑓,𝑔,𝑘)   𝑍(𝑣,𝑔)

Proof of Theorem upeu2

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		upeu2.n	. . 3 ⊢ (𝜑 → 𝑁 = (((𝑋𝐺𝑌)‘𝐾)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
2		upcic.c	. . . 4 ⊢ 𝐶 = (Base‘𝐸)
3		upcic.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐸)
4		upcic.o	. . . 4 ⊢ 𝑂 = (comp‘𝐸)
5		upcic.f	. . . . 5 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
6	5	funcrcl3 49555	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
7		upcic.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐶)
8		upcic.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐷)
9	8, 2, 5	funcf1 17833	. . . . 5 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
10		upcic.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11	9, 10	ffvelcdmd 7037	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐶)
12		upcic.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
13	9, 12	ffvelcdmd 7037	. . . 4 ⊢ (𝜑 → (𝐹‘𝑌) ∈ 𝐶)
14		upcic.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑍𝐽(𝐹‘𝑋)))
15		upcic.h	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝐷)
16	8, 15, 3, 5, 10, 12	funcf2 17835	. . . . 5 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
17		upeu2.i	. . . . . . 7 ⊢ 𝐼 = (Iso‘𝐷)
18	5	funcrcl2 49554	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
19	8, 15, 17, 18, 10, 12	isohom 17743	. . . . . 6 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋𝐻𝑌))
20		upeu2.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐼𝑌))
21	19, 20	sseldd 3922	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))
22	16, 21	ffvelcdmd 7037	. . . 4 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝐾) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
23	2, 3, 4, 6, 7, 11, 13, 14, 22	catcocl 17651	. . 3 ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝐾)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) ∈ (𝑍𝐽(𝐹‘𝑌)))
24	1, 23	eqeltrd 2836	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑍𝐽(𝐹‘𝑌)))
25		upcic.1	. . . . . 6 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ∀𝑓 ∈ (𝑍𝐽(𝐹‘𝑤))∃!𝑘 ∈ (𝑋𝐻𝑤)𝑓 = (((𝑋𝐺𝑤)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑤))𝑀))
26	25	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) → ∀𝑤 ∈ 𝐵 ∀𝑓 ∈ (𝑍𝐽(𝐹‘𝑤))∃!𝑘 ∈ (𝑋𝐻𝑤)𝑓 = (((𝑋𝐺𝑤)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑤))𝑀))
27		simprl 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) → 𝑣 ∈ 𝐵)
28		simprr 773	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) → 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))
29	26, 27, 28	upciclem1 49641	. . . 4 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) → ∃!𝑝 ∈ (𝑋𝐻𝑣)𝑔 = (((𝑋𝐺𝑣)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑣))𝑀))
30		eqid 2736	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
31	18	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑙 ∈ (𝑌𝐻𝑣)) → 𝐷 ∈ Cat)
32	10	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑙 ∈ (𝑌𝐻𝑣)) → 𝑋 ∈ 𝐵)
33	12	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑙 ∈ (𝑌𝐻𝑣)) → 𝑌 ∈ 𝐵)
34	27	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑙 ∈ (𝑌𝐻𝑣)) → 𝑣 ∈ 𝐵)
35	21	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑙 ∈ (𝑌𝐻𝑣)) → 𝐾 ∈ (𝑋𝐻𝑌))
36		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑙 ∈ (𝑌𝐻𝑣)) → 𝑙 ∈ (𝑌𝐻𝑣))
37	8, 15, 30, 31, 32, 33, 34, 35, 36	catcocl 17651	. . . . 5 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑙 ∈ (𝑌𝐻𝑣)) → (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾) ∈ (𝑋𝐻𝑣))
38	18	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑝 ∈ (𝑋𝐻𝑣)) → 𝐷 ∈ Cat)
39	10	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑝 ∈ (𝑋𝐻𝑣)) → 𝑋 ∈ 𝐵)
40	12	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑝 ∈ (𝑋𝐻𝑣)) → 𝑌 ∈ 𝐵)
41	27	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑝 ∈ (𝑋𝐻𝑣)) → 𝑣 ∈ 𝐵)
42	20	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑝 ∈ (𝑋𝐻𝑣)) → 𝐾 ∈ (𝑋𝐼𝑌))
43		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑝 ∈ (𝑋𝐻𝑣)) → 𝑝 ∈ (𝑋𝐻𝑣))
44	8, 15, 30, 17, 38, 39, 40, 41, 42, 43	upeu2lem 49503	. . . . 5 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑝 ∈ (𝑋𝐻𝑣)) → ∃!𝑙 ∈ (𝑌𝐻𝑣)𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))
45		simprr 773	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))
46	45	fveq2d 6844	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → ((𝑋𝐺𝑣)‘𝑝) = ((𝑋𝐺𝑣)‘(𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾)))
47	46	oveq1d 7382	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → (((𝑋𝐺𝑣)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑣))𝑀) = (((𝑋𝐺𝑣)‘(𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑣))𝑀))
48	5	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → 𝐹(𝐷 Func 𝐸)𝐺)
49	10	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → 𝑋 ∈ 𝐵)
50	12	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → 𝑌 ∈ 𝐵)
51	27	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → 𝑣 ∈ 𝐵)
52	7	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → 𝑍 ∈ 𝐶)
53	14	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → 𝑀 ∈ (𝑍𝐽(𝐹‘𝑋)))
54	21	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → 𝐾 ∈ (𝑋𝐻𝑌))
55		simprl 771	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → 𝑙 ∈ (𝑌𝐻𝑣))
56	1	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → 𝑁 = (((𝑋𝐺𝑌)‘𝐾)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
57	8, 2, 15, 3, 4, 48, 49, 50, 51, 52, 53, 30, 54, 55, 56	upciclem2 49642	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → (((𝑋𝐺𝑣)‘(𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑣))𝑀) = (((𝑌𝐺𝑣)‘𝑙)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑣))𝑁))
58	47, 57	eqtrd 2771	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → (((𝑋𝐺𝑣)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑣))𝑀) = (((𝑌𝐺𝑣)‘𝑙)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑣))𝑁))
59	58	eqeq2d 2747	. . . . 5 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → (𝑔 = (((𝑋𝐺𝑣)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑣))𝑀) ↔ 𝑔 = (((𝑌𝐺𝑣)‘𝑙)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑣))𝑁)))
60	37, 44, 59	reuxfr1dd 49282	. . . 4 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) → (∃!𝑝 ∈ (𝑋𝐻𝑣)𝑔 = (((𝑋𝐺𝑣)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑣))𝑀) ↔ ∃!𝑙 ∈ (𝑌𝐻𝑣)𝑔 = (((𝑌𝐺𝑣)‘𝑙)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑣))𝑁)))
61	29, 60	mpbid 232	. . 3 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) → ∃!𝑙 ∈ (𝑌𝐻𝑣)𝑔 = (((𝑌𝐺𝑣)‘𝑙)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑣))𝑁))
62	61	ralrimivva 3180	. 2 ⊢ (𝜑 → ∀𝑣 ∈ 𝐵 ∀𝑔 ∈ (𝑍𝐽(𝐹‘𝑣))∃!𝑙 ∈ (𝑌𝐻𝑣)𝑔 = (((𝑌𝐺𝑣)‘𝑙)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑣))𝑁))
63	24, 62	jca 511	1 ⊢ (𝜑 → (𝑁 ∈ (𝑍𝐽(𝐹‘𝑌)) ∧ ∀𝑣 ∈ 𝐵 ∀𝑔 ∈ (𝑍𝐽(𝐹‘𝑣))∃!𝑙 ∈ (𝑌𝐻𝑣)𝑔 = (((𝑌𝐺𝑣)‘𝑙)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑣))𝑁)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃!wreu 3340  ⟨cop 4573   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  Hom chom 17231  compcco 17232  Catccat 17630  Isociso 17713   Func cfunc 17821

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775  df-ixp 8846  df-cat 17634  df-cid 17635  df-sect 17714  df-inv 17715  df-iso 17716  df-func 17825

This theorem is referenced by:  upeu4  49671