Theorem upeu2 49950

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 49858	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
7		upcic.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐶)
8		upcic.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐷)
9	8, 2, 5	funcf1 17918	. . . . 5 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
10		upcic.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11	9, 10	ffvelcdmd 7080	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐶)
12		upcic.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
13	9, 12	ffvelcdmd 7080	. . . 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 17920	. . . . 5 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
17		upeu2.i	. . . . . . 7 ⊢ 𝐼 = (Iso‘𝐷)
18	5	funcrcl2 49857	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
19	8, 15, 17, 18, 10, 12	isohom 17828	. . . . . 6 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋𝐻𝑌))
20		upeu2.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐼𝑌))
21	19, 20	sseldd 3938	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))
22	16, 21	ffvelcdmd 7080	. . . 4 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝐾) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
23	2, 3, 4, 6, 7, 11, 13, 14, 22	catcocl 17736	. . 3 ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝐾)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) ∈ (𝑍𝐽(𝐹‘𝑌)))
24	1, 23	eqeltrd 2863	. 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 485	. . . . 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 782	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) → 𝑣 ∈ 𝐵)
28		simprr 784	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) → 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))
29	26, 27, 28	upciclem1 49944	. . . 4 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) → ∃!𝑝 ∈ (𝑋𝐻𝑣)𝑔 = (((𝑋𝐺𝑣)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑣))𝑀))
30		eqid 2763	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
31	18	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑙 ∈ (𝑌𝐻𝑣)) → 𝐷 ∈ Cat)
32	10	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑙 ∈ (𝑌𝐻𝑣)) → 𝑋 ∈ 𝐵)
33	12	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑙 ∈ (𝑌𝐻𝑣)) → 𝑌 ∈ 𝐵)
34	27	adantr 485	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑙 ∈ (𝑌𝐻𝑣)) → 𝑣 ∈ 𝐵)
35	21	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑙 ∈ (𝑌𝐻𝑣)) → 𝐾 ∈ (𝑋𝐻𝑌))
36		simpr 489	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑙 ∈ (𝑌𝐻𝑣)) → 𝑙 ∈ (𝑌𝐻𝑣))
37	8, 15, 30, 31, 32, 33, 34, 35, 36	catcocl 17736	. . . . 5 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑙 ∈ (𝑌𝐻𝑣)) → (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾) ∈ (𝑋𝐻𝑣))
38	18	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑝 ∈ (𝑋𝐻𝑣)) → 𝐷 ∈ Cat)
39	10	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑝 ∈ (𝑋𝐻𝑣)) → 𝑋 ∈ 𝐵)
40	12	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑝 ∈ (𝑋𝐻𝑣)) → 𝑌 ∈ 𝐵)
41	27	adantr 485	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑝 ∈ (𝑋𝐻𝑣)) → 𝑣 ∈ 𝐵)
42	20	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑝 ∈ (𝑋𝐻𝑣)) → 𝐾 ∈ (𝑋𝐼𝑌))
43		simpr 489	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑝 ∈ (𝑋𝐻𝑣)) → 𝑝 ∈ (𝑋𝐻𝑣))
44	8, 15, 30, 17, 38, 39, 40, 41, 42, 43	upeu2lem 49806	. . . . 5 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑝 ∈ (𝑋𝐻𝑣)) → ∃!𝑙 ∈ (𝑌𝐻𝑣)𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))
45		simprr 784	. . . . . . . . 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 6885	. . . . . . . 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 7425	. . . . . . 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 738	. . . . . . . 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 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → 𝑋 ∈ 𝐵)
50	12	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → 𝑌 ∈ 𝐵)
51	27	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → 𝑣 ∈ 𝐵)
52	7	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → 𝑍 ∈ 𝐶)
53	14	ad2antrr 738	. . . . . . . 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 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → 𝐾 ∈ (𝑋𝐻𝑌))
55		simprl 782	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → 𝑙 ∈ (𝑌𝐻𝑣))
56	1	ad2antrr 738	. . . . . . . 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 49945	. . . . . . 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 2798	. . . . . 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 2774	. . . . 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 49585	. . . 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 235	. . 3 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) → ∃!𝑙 ∈ (𝑌𝐻𝑣)𝑔 = (((𝑌𝐺𝑣)‘𝑙)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑣))𝑁))
62	61	ralrimivva 3208	. 2 ⊢ (𝜑 → ∀𝑣 ∈ 𝐵 ∀𝑔 ∈ (𝑍𝐽(𝐹‘𝑣))∃!𝑙 ∈ (𝑌𝐻𝑣)𝑔 = (((𝑌𝐺𝑣)‘𝑙)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑣))𝑁))
63	24, 62	jca 520	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 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃!wreu 3367  ⟨cop 4595   class class class wbr 5109  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  Hom chom 17316  compcco 17317  Catccat 17715  Isociso 17798   Func cfunc 17906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-map 8822  df-ixp 8892  df-cat 17719  df-cid 17720  df-sect 17799  df-inv 17800  df-iso 17801  df-func 17910

This theorem is referenced by:  upeu4  49974