Theorem upeu2 49417

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 49325	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
7		upcic.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐶)
8		upcic.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐷)
9	8, 2, 5	funcf1 17790	. . . . 5 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
10		upcic.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11	9, 10	ffvelcdmd 7030	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐶)
12		upcic.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
13	9, 12	ffvelcdmd 7030	. . . 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 17792	. . . . 5 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
17		upeu2.i	. . . . . . 7 ⊢ 𝐼 = (Iso‘𝐷)
18	5	funcrcl2 49324	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
19	8, 15, 17, 18, 10, 12	isohom 17700	. . . . . 6 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋𝐻𝑌))
20		upeu2.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐼𝑌))
21	19, 20	sseldd 3934	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))
22	16, 21	ffvelcdmd 7030	. . . 4 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝐾) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
23	2, 3, 4, 6, 7, 11, 13, 14, 22	catcocl 17608	. . 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 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) → 𝑣 ∈ 𝐵)
28		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) → 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))
29	26, 27, 28	upciclem1 49411	. . . 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 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑙 ∈ (𝑌𝐻𝑣)) → 𝐷 ∈ Cat)
32	10	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑙 ∈ (𝑌𝐻𝑣)) → 𝑋 ∈ 𝐵)
33	12	ad2antrr 726	. . . . . 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 726	. . . . . 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 17608	. . . . 5 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑙 ∈ (𝑌𝐻𝑣)) → (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾) ∈ (𝑋𝐻𝑣))
38	18	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑝 ∈ (𝑋𝐻𝑣)) → 𝐷 ∈ Cat)
39	10	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑝 ∈ (𝑋𝐻𝑣)) → 𝑋 ∈ 𝐵)
40	12	ad2antrr 726	. . . . . 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 726	. . . . . 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 49273	. . . . 5 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑝 ∈ (𝑋𝐻𝑣)) → ∃!𝑙 ∈ (𝑌𝐻𝑣)𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))
45		simprr 772	. . . . . . . . 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 6838	. . . . . . . 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 7373	. . . . . . 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 726	. . . . . . . 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 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → 𝑋 ∈ 𝐵)
50	12	ad2antrr 726	. . . . . . . 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 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → 𝑍 ∈ 𝐶)
53	14	ad2antrr 726	. . . . . . . 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 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → 𝐾 ∈ (𝑋𝐻𝑌))
55		simprl 770	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → 𝑙 ∈ (𝑌𝐻𝑣))
56	1	ad2antrr 726	. . . . . . . 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 49412	. . . . . . 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 49052	. . . 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 3179	. 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 1541   ∈ wcel 2113  ∀wral 3051  ∃!wreu 3348  ⟨cop 4586   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  Hom chom 17188  compcco 17189  Catccat 17587  Isociso 17670   Func cfunc 17778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8765  df-ixp 8836  df-cat 17591  df-cid 17592  df-sect 17671  df-inv 17672  df-iso 17673  df-func 17782

This theorem is referenced by:  upeu4  49441