Theorem upciclem4 49751

Description: Lemma for upcic 49752 and upeu 49753. (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 49748	. . . 4 ⊢ (𝜑 → ∃!𝑝 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
5		reurex 3370	. . . 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 17	. . 3 ⊢ (𝜑 → ∃𝑝 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
7		simpl 486	. . . . 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 49748	. . . . 5 ⊢ (𝜑 → ∃!𝑞 ∈ (𝑌𝐻𝑋)𝑀 = (((𝑌𝐺𝑋)‘𝑞)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))
12		reurex 3370	. . . . 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 18	. . . 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 2761	. . . . 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 736	. . . . . 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 49661	. . . . 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 736	. . . . 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 736	. . . . 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 2761	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
23		eqid 2761	. . . . . 6 ⊢ (Id‘𝐷) = (Id‘𝐷)
24		simplrl 786	. . . . . 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 780	. . . . . 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 736	. . . . . . 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 736	. . . . . . 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 736	. . . . . . 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 782	. . . . . . 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 787	. . . . . . 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 49750	. . . . . 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 736	. . . . . . 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 736	. . . . . . 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 49750	. . . . . 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 49609	. . . . 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 17824	. . . 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 3168	. . 3 ⊢ ((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) → 𝑋( ≃𝑐 ‘𝐷)𝑌)
42	6, 41	rexlimddv 3168	. 2 ⊢ (𝜑 → 𝑋( ≃𝑐 ‘𝐷)𝑌)
43	13, 39	rexlimddv 3168	. . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ (𝑋𝐻𝑌) ∧ 𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) → 𝑝 ∈ (𝑋(Iso‘𝐷)𝑌))
44		simprr 782	. . . 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 3179	. . 3 ⊢ (𝜑 → ∃𝑝 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
46		fveq2 6862	. . . . . 6 ⊢ (𝑝 = 𝑟 → ((𝑋𝐺𝑌)‘𝑝) = ((𝑋𝐺𝑌)‘𝑟))
47	46	oveq1d 7406	. . . . 5 ⊢ (𝑝 = 𝑟 → (((𝑋𝐺𝑌)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
48	47	eqeq2d 2772	. . . 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 3240	. . 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 220	. 2 ⊢ (𝜑 → ∃𝑟 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
51	42, 50	jca 519	1 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐷)𝑌 ∧ ∃𝑟 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085  ∃!wreu 3364  ⟨cop 4585   class class class wbr 5097  ‘cfv 6516  (class class class)co 7391  Basecbs 17236  Hom chom 17288  compcco 17289  Idccid 17688  Isociso 17770   ≃_𝑐 ccic 17819   Func cfunc 17878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-1st 7965  df-2nd 7966  df-supp 8135  df-map 8804  df-ixp 8874  df-cat 17691  df-cid 17692  df-sect 17771  df-inv 17772  df-iso 17773  df-cic 17820  df-func 17882

This theorem is referenced by:  upcic  49752  upeu  49753