Theorem upciclem4 50616

Description: Lemma for upcic 50617 and upeu 50618. (Contributed by Zhi Wang, 19-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))
upcic.n	⊢ (𝜑 → 𝑁 ∈ (𝑍𝐽(𝐹‘𝑌)))
upcic.2	⊢ (𝜑 → ∀𝑣 ∈ 𝐵 ∀𝑔 ∈ (𝑍𝐽(𝐹‘𝑣))∃!𝑙 ∈ (𝑌𝐻𝑣)𝑔 = (((𝑌𝐺𝑣)‘𝑙)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑣))𝑁))

Assertion

Ref	Expression
upciclem4	⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐷)𝑌 ∧ ∃𝑟 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)))

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

Allowed substitution hints:   𝜑(𝑤, 𝑣, 𝑓, 𝑔, 𝑘, 𝑟, 𝑙)   𝐵(𝑓, 𝑔, 𝑘, 𝑟, 𝑙)   𝐶(𝑤, 𝑣, 𝑓, 𝑔, 𝑘, 𝑟, 𝑙)   𝐷(𝑤, 𝑣, 𝑓, 𝑔, 𝑘, 𝑙)   𝐸(𝑤, 𝑣, 𝑓, 𝑔, 𝑘, 𝑟, 𝑙)   𝐽(𝑘, 𝑟, 𝑙)   𝑀(𝑣)   𝑁(𝑤)

Proof of Theorem upciclem4

Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		upcic.1	. . . . 5 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ∀𝑓 ∈ (𝑍𝐽(𝐹‘𝑤))∃!𝑘 ∈ (𝑋𝐻𝑤)𝑓 = (((𝑋𝐺𝑤)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑤))𝑀))
2		upcic.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
3		upcic.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝑍𝐽(𝐹‘𝑌)))
4	1, 2, 3	upciclem1 50613	. . . 4 ⊢ (𝜑 → ∃!𝑝 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
5		reurex 3373	. . . 4 ⊢ (∃!𝑝 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) → ∃𝑝 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
6	4, 5	syl 18	. . 3 ⊢ (𝜑 → ∃𝑝 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
7		simpl 488	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) → 𝜑)
8		upcic.2	. . . . . 6 ⊢ (𝜑 → ∀𝑣 ∈ 𝐵 ∀𝑔 ∈ (𝑍𝐽(𝐹‘𝑣))∃!𝑙 ∈ (𝑌𝐻𝑣)𝑔 = (((𝑌𝐺𝑣)‘𝑙)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑣))𝑁))
9		upcic.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10		upcic.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (𝑍𝐽(𝐹‘𝑋)))
11	8, 9, 10	upciclem1 50613	. . . . 5 ⊢ (𝜑 → ∃!𝑞 ∈ (𝑌𝐻𝑋)𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))
12		reurex 3373	. . . . 5 ⊢ (∃!𝑞 ∈ (𝑌𝐻𝑋)𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁) → ∃𝑞 ∈ (𝑌𝐻𝑋)𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))
13	7, 11, 12	3syl 19	. . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) → ∃𝑞 ∈ (𝑌𝐻𝑋)𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))
14		eqid 2763	. . . . 5 ⊢ (Iso‘𝐷) = (Iso‘𝐷)
15		upcic.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐷)
16		upcic.f	. . . . . . 7 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
17	16	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) ∧ (𝑞 ∈ (𝑌𝐻𝑋) ∧ 𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))) → 𝐹(𝐷 Func 𝐸)𝐺)
18	17	funcrcl2 50524	. . . . 5 ⊢ (((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) ∧ (𝑞 ∈ (𝑌𝐻𝑋) ∧ 𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))) → 𝐷 ∈ Cat)
19	9	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) ∧ (𝑞 ∈ (𝑌𝐻𝑋) ∧ 𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))) → 𝑋 ∈ 𝐵)
20	2	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) ∧ (𝑞 ∈ (𝑌𝐻𝑋) ∧ 𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))) → 𝑌 ∈ 𝐵)
21		upcic.h	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝐷)
22		eqid 2763	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
23		eqid 2763	. . . . . 6 ⊢ (Id‘𝐷) = (Id‘𝐷)
24		simplrl 789	. . . . . 6 ⊢ (((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) ∧ (𝑞 ∈ (𝑌𝐻𝑋) ∧ 𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))) → 𝑝 ∈ (𝑋𝐻𝑌))
25		simprl 783	. . . . . 6 ⊢ (((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) ∧ (𝑞 ∈ (𝑌𝐻𝑋) ∧ 𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))) → 𝑞 ∈ (𝑌𝐻𝑋))
26		upcic.c	. . . . . . 7 ⊢ 𝐶 = (Base‘𝐸)
27		upcic.j	. . . . . . 7 ⊢ 𝐽 = (Hom ‘𝐸)
28		upcic.o	. . . . . . 7 ⊢ 𝑂 = (comp‘𝐸)
29		upcic.z	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ 𝐶)
30	29	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) ∧ (𝑞 ∈ (𝑌𝐻𝑋) ∧ 𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))) → 𝑍 ∈ 𝐶)
31	10	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) ∧ (𝑞 ∈ (𝑌𝐻𝑋) ∧ 𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))) → 𝑀 ∈ (𝑍𝐽(𝐹‘𝑋)))
32	1	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) ∧ (𝑞 ∈ (𝑌𝐻𝑋) ∧ 𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))) → ∀𝑤 ∈ 𝐵 ∀𝑓 ∈ (𝑍𝐽(𝐹‘𝑤))∃!𝑘 ∈ (𝑋𝐻𝑤)𝑓 = (((𝑋𝐺𝑤)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑤))𝑀))
33		simprr 785	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) ∧ (𝑞 ∈ (𝑌𝐻𝑋) ∧ 𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))) → 𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))
34		simplrr 790	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) ∧ (𝑞 ∈ (𝑌𝐻𝑋) ∧ 𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))) → 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
35	15, 26, 21, 27, 28, 17, 19, 20, 30, 31, 32, 22, 24, 25, 33, 34	upciclem3 50615	. . . . . 6 ⊢ (((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) ∧ (𝑞 ∈ (𝑌𝐻𝑋) ∧ 𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))) → (𝑞(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑋)𝑝) = ((Id‘𝐷)‘𝑋))
36	3	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) ∧ (𝑞 ∈ (𝑌𝐻𝑋) ∧ 𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))) → 𝑁 ∈ (𝑍𝐽(𝐹‘𝑌)))
37	8	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) ∧ (𝑞 ∈ (𝑌𝐻𝑋) ∧ 𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))) → ∀𝑣 ∈ 𝐵 ∀𝑔 ∈ (𝑍𝐽(𝐹‘𝑣))∃!𝑙 ∈ (𝑌𝐻𝑣)𝑔 = (((𝑌𝐺𝑣)‘𝑙)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑣))𝑁))
38	15, 26, 21, 27, 28, 17, 20, 19, 30, 36, 37, 22, 25, 24, 34, 33	upciclem3 50615	. . . . . 6 ⊢ (((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) ∧ (𝑞 ∈ (𝑌𝐻𝑋) ∧ 𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))) → (𝑝(⟨𝑌, 𝑋⟩(comp‘𝐷)𝑌)𝑞) = ((Id‘𝐷)‘𝑌))
39	15, 21, 22, 14, 23, 18, 19, 20, 24, 25, 35, 38	isisod 50472	. . . . 5 ⊢ (((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) ∧ (𝑞 ∈ (𝑌𝐻𝑋) ∧ 𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))) → 𝑝 ∈ (𝑋(Iso‘𝐷)𝑌))
40	14, 15, 18, 19, 20, 39	brcici 18025	. . . 4 ⊢ (((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) ∧ (𝑞 ∈ (𝑌𝐻𝑋) ∧ 𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))) → 𝑋( ≃𝑐 ‘𝐷)𝑌)
41	13, 40	rexlimddv 3173	. . 3 ⊢ ((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) → 𝑋( ≃𝑐 ‘𝐷)𝑌)
42	6, 41	rexlimddv 3173	. 2 ⊢ (𝜑 → 𝑋( ≃𝑐 ‘𝐷)𝑌)
43	13, 39	rexlimddv 3173	. . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) → 𝑝 ∈ (𝑋(Iso‘𝐷)𝑌))
44		simprr 785	. . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) → 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
45	6, 43, 44	reximssdv 3184	. . 3 ⊢ (𝜑 → ∃𝑝 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
46		fveq2 6916	. . . . . 6 ⊢ (𝑝 = 𝑟 → ((𝑋𝐺𝑌)‘𝑝) = ((𝑋𝐺𝑌)‘𝑟))
47	46	oveq1d 7466	. . . . 5 ⊢ (𝑝 = 𝑟 → (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
48	47	eqeq2d 2774	. . . 4 ⊢ (𝑝 = 𝑟 → (𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) ↔ 𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)))
49	48	cbvrexvw 3245	. . 3 ⊢ (∃𝑝 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) ↔ ∃𝑟 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
50	45, 49	sylib 221	. 2 ⊢ (𝜑 → ∃𝑟 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
51	42, 50	jca 521	1 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐷)𝑌 ∧ ∃𝑟 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  ∃!wreu 3367  ⟨cop 4840   class class class wbr 5108  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  Hom chom 17483  compcco 17484  Idccid 17889  Isociso 17971   ≃_𝑐 ccic 18020   Func cfunc 18079

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-supp 8215  df-map 8887  df-ixp 8964  df-cat 17892  df-cid 17893  df-sect 17972  df-inv 17973  df-iso 17974  df-cic 18021  df-func 18083

This theorem is used by:  upcic  50617  upeu  50618