Theorem upeu2 48782

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 48774	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
7		upcic.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐶)
8		upcic.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐷)
9	8, 2, 5	funcf1 17951	. . . . 5 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
10		upcic.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11	9, 10	ffvelcdmd 7123	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐶)
12		upcic.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
13	9, 12	ffvelcdmd 7123	. . . 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 17953	. . . . 5 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
17		upeu2.i	. . . . . . 7 ⊢ 𝐼 = (Iso‘𝐷)
18	5	funcrcl2 48773	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
19	8, 15, 17, 18, 10, 12	isohom 17858	. . . . . 6 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋𝐻𝑌))
20		upeu2.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐼𝑌))
21	19, 20	sseldd 4010	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))
22	16, 21	ffvelcdmd 7123	. . . 4 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝐾) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
23	2, 3, 4, 6, 7, 11, 13, 14, 22	catcocl 17764	. . 3 ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝐾)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) ∈ (𝑍𝐽(𝐹‘𝑌)))
24	1, 23	eqeltrd 2844	. 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 48776	. . . 4 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) → ∃!𝑝 ∈ (𝑋𝐻𝑣)𝑔 = (((𝑋𝐺𝑣)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑣))𝑀))
30		eqid 2740	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
31	18	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑙 ∈ (𝑌𝐻𝑣)) → 𝐷 ∈ Cat)
32	10	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑙 ∈ (𝑌𝐻𝑣)) → 𝑋 ∈ 𝐵)
33	12	ad2antrr 725	. . . . . 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 725	. . . . . 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 17764	. . . . 5 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑙 ∈ (𝑌𝐻𝑣)) → (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾) ∈ (𝑋𝐻𝑣))
38	18	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑝 ∈ (𝑋𝐻𝑣)) → 𝐷 ∈ Cat)
39	10	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ 𝑝 ∈ (𝑋𝐻𝑣)) → 𝑋 ∈ 𝐵)
40	12	ad2antrr 725	. . . . . 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 725	. . . . . 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 48772	. . . . 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 6928	. . . . . . . 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 7467	. . . . . . 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 725	. . . . . . . 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 725	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → 𝑋 ∈ 𝐵)
50	12	ad2antrr 725	. . . . . . . 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 725	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑔 ∈ (𝑍𝐽(𝐹‘𝑣)))) ∧ (𝑙 ∈ (𝑌𝐻𝑣) ∧ 𝑝 = (𝑙(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑣)𝐾))) → 𝑍 ∈ 𝐶)
53	14	ad2antrr 725	. . . . . . . 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 725	. . . . . . . 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 725	. . . . . . . 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 48777	. . . . . . 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 2780	. . . . . 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 2751	. . . . 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 48619	. . . 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 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 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 1537   ∈ wcel 2108  ∀wral 3067  ∃!wreu 3386  ⟨cop 4655   class class class wbr 5168  ‘cfv 6577  (class class class)co 7452  Basecbs 17279  Hom chom 17343  compcco 17344  Catccat 17743  Isociso 17828   Func cfunc 17939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5305  ax-sep 5319  ax-nul 5326  ax-pow 5385  ax-pr 5449  ax-un 7774

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3445  df-v 3491  df-sbc 3806  df-csb 3923  df-dif 3980  df-un 3982  df-in 3984  df-ss 3994  df-nul 4354  df-if 4550  df-pw 4625  df-sn 4650  df-pr 4652  df-op 4656  df-uni 4934  df-iun 5019  df-br 5169  df-opab 5231  df-mpt 5252  df-id 5595  df-xp 5708  df-rel 5709  df-cnv 5710  df-co 5711  df-dm 5712  df-rn 5713  df-res 5714  df-ima 5715  df-iota 6529  df-fun 6579  df-fn 6580  df-f 6581  df-f1 6582  df-fo 6583  df-f1o 6584  df-fv 6585  df-riota 7408  df-ov 7455  df-oprab 7456  df-mpo 7457  df-1st 8034  df-2nd 8035  df-map 8890  df-ixp 8960  df-cat 17747  df-cid 17748  df-sect 17829  df-inv 17830  df-iso 17831  df-func 17943

This theorem is referenced by: (None)