Theorem upeu2 49531

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 49439	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
7		upcic.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐶)
8		upcic.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐷)
9	8, 2, 5	funcf1 17802	. . . . 5 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
10		upcic.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11	9, 10	ffvelcdmd 7039	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐶)
12		upcic.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
13	9, 12	ffvelcdmd 7039	. . . 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 17804	. . . . 5 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
17		upeu2.i	. . . . . . 7 ⊢ 𝐼 = (Iso‘𝐷)
18	5	funcrcl2 49438	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
19	8, 15, 17, 18, 10, 12	isohom 17712	. . . . . 6 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋𝐻𝑌))
20		upeu2.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐼𝑌))
21	19, 20	sseldd 3936	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))
22	16, 21	ffvelcdmd 7039	. . . 4 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝐾) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
23	2, 3, 4, 6, 7, 11, 13, 14, 22	catcocl 17620	. . 3 ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝐾)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) ∈ (𝑍𝐽(𝐹‘𝑌)))
24	1, 23	eqeltrd 2837	. 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 49525	. . . 4 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) → ∃!𝑝 ∈ (𝑋𝐻𝑣)𝑔 = (((𝑋𝐺𝑣)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑣))𝑀))
30		eqid 2737	. . . . . 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 17620	. . . . 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 49387	. . . . 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 6846	. . . . . . . 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 7383	. . . . . . 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 49526	. . . . . . 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 2772	. . . . . 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 2748	. . . . 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 49166	. . . 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 3181	. 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 3052  ∃!wreu 3350  ⟨cop 4588   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  Hom chom 17200  compcco 17201  Catccat 17599  Isociso 17682   Func cfunc 17790

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 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-map 8777  df-ixp 8848  df-cat 17603  df-cid 17604  df-sect 17683  df-inv 17684  df-iso 17685  df-func 17794

This theorem is referenced by:  upeu4  49555