Theorem imaf1co 49342

Description: An image of a functor whose object part is injective preserves the composition. (Contributed by Zhi Wang, 7-Nov-2025.)

Hypotheses

Ref	Expression
imasubc.s	⊢ 𝑆 = (𝐹 “ 𝐴)
imasubc.h	⊢ 𝐻 = (Hom ‘𝐷)
imasubc.k	⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
imassc.f	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
imaf1co.b	⊢ 𝐵 = (Base‘𝐷)
imaf1co.c	⊢ 𝐶 = (Base‘𝐸)
imaf1co.o	⊢ ∙ = (comp‘𝐸)
imaf1co.f	⊢ (𝜑 → 𝐹:𝐵–1-1→𝐶)
imaf1co.x	⊢ (𝜑 → 𝑋 ∈ 𝑆)
imaf1co.y	⊢ (𝜑 → 𝑌 ∈ 𝑆)
imaf1co.z	⊢ (𝜑 → 𝑍 ∈ 𝑆)
imaf1co.m	⊢ (𝜑 → 𝑀 ∈ (𝑋𝐾𝑌))
imaf1co.n	⊢ (𝜑 → 𝑁 ∈ (𝑌𝐾𝑍))

Assertion

Ref	Expression
imaf1co	⊢ (𝜑 → (𝑁(⟨𝑋, 𝑌⟩ ∙ 𝑍)𝑀) ∈ (𝑋𝐾𝑍))

Distinct variable groups:   𝐹,𝑝,𝑥,𝑦   𝐺,𝑝,𝑥,𝑦   𝐻,𝑝,𝑥,𝑦   𝑥,𝑆,𝑦   𝑋,𝑝,𝑥,𝑦   𝑌,𝑝,𝑥,𝑦   𝑍,𝑝,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑝)   𝐴(𝑥,𝑦,𝑝)   𝐵(𝑥,𝑦,𝑝)   𝐶(𝑥,𝑦,𝑝)   𝐷(𝑥,𝑦,𝑝)   𝑆(𝑝)   ∙ (𝑥,𝑦,𝑝)   𝐸(𝑥,𝑦,𝑝)   𝐾(𝑥,𝑦,𝑝)   𝑀(𝑥,𝑦,𝑝)   𝑁(𝑥,𝑦,𝑝)

Proof of Theorem imaf1co

Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imaf1co.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐷)
2		imasubc.h	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝐷)
3		eqid 2734	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
4		imassc.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
5	4	funcrcl2 49266	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
6	5	ad4antr 732	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) → 𝐷 ∈ Cat)
7		imasubc.s	. . . . . . . . 9 ⊢ 𝑆 = (𝐹 “ 𝐴)
8		imaf1co.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐵–1-1→𝐶)
9		imaf1co.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
10	7, 8, 9	imaf1homlem 49294	. . . . . . . 8 ⊢ (𝜑 → ({(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}) ∧ (𝐹‘(◡𝐹‘𝑋)) = 𝑋 ∧ (◡𝐹‘𝑋) ∈ 𝐵))
11	10	simp3d 1144	. . . . . . 7 ⊢ (𝜑 → (◡𝐹‘𝑋) ∈ 𝐵)
12	11	ad4antr 732	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) → (◡𝐹‘𝑋) ∈ 𝐵)
13		imaf1co.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑆)
14	7, 8, 13	imaf1homlem 49294	. . . . . . . 8 ⊢ (𝜑 → ({(◡𝐹‘𝑌)} = (◡𝐹 “ {𝑌}) ∧ (𝐹‘(◡𝐹‘𝑌)) = 𝑌 ∧ (◡𝐹‘𝑌) ∈ 𝐵))
15	14	simp3d 1144	. . . . . . 7 ⊢ (𝜑 → (◡𝐹‘𝑌) ∈ 𝐵)
16	15	ad4antr 732	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) → (◡𝐹‘𝑌) ∈ 𝐵)
17		imaf1co.z	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ 𝑆)
18	7, 8, 17	imaf1homlem 49294	. . . . . . . 8 ⊢ (𝜑 → ({(◡𝐹‘𝑍)} = (◡𝐹 “ {𝑍}) ∧ (𝐹‘(◡𝐹‘𝑍)) = 𝑍 ∧ (◡𝐹‘𝑍) ∈ 𝐵))
19	18	simp3d 1144	. . . . . . 7 ⊢ (𝜑 → (◡𝐹‘𝑍) ∈ 𝐵)
20	19	ad4antr 732	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) → (◡𝐹‘𝑍) ∈ 𝐵)
21		simp-4r 783	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) → 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌)))
22		simplr 768	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) → 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍)))
23	1, 2, 3, 6, 12, 16, 20, 21, 22	catcocl 17606	. . . . 5 ⊢ (((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) → (𝑛(⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩(comp‘𝐷)(◡𝐹‘𝑍))𝑚) ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑍)))
24		eqid 2734	. . . . . . . 8 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
25	1, 2, 24, 4, 11, 19	funcf2 17790	. . . . . . 7 ⊢ (𝜑 → ((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑍)):((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑍))⟶((𝐹‘(◡𝐹‘𝑋))(Hom ‘𝐸)(𝐹‘(◡𝐹‘𝑍))))
26	25	ad4antr 732	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) → ((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑍)):((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑍))⟶((𝐹‘(◡𝐹‘𝑋))(Hom ‘𝐸)(𝐹‘(◡𝐹‘𝑍))))
27	26	funfvima2d 7176	. . . . 5 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) ∧ (𝑛(⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩(comp‘𝐷)(◡𝐹‘𝑍))𝑚) ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑍))) → (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑍))‘(𝑛(⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩(comp‘𝐷)(◡𝐹‘𝑍))𝑚)) ∈ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑍)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑍))))
28	23, 27	mpdan 687	. . . 4 ⊢ (((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) → (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑍))‘(𝑛(⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩(comp‘𝐷)(◡𝐹‘𝑍))𝑚)) ∈ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑍)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑍))))
29		imaf1co.o	. . . . . 6 ⊢ ∙ = (comp‘𝐸)
30	4	ad4antr 732	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) → 𝐹(𝐷 Func 𝐸)𝐺)
31	1, 2, 3, 29, 30, 12, 16, 20, 21, 22	funcco 17793	. . . . 5 ⊢ (((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) → (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑍))‘(𝑛(⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩(comp‘𝐷)(◡𝐹‘𝑍))𝑚)) = ((((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛)(⟨(𝐹‘(◡𝐹‘𝑋)), (𝐹‘(◡𝐹‘𝑌))⟩ ∙ (𝐹‘(◡𝐹‘𝑍)))(((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚)))
32	10	simp2d 1143	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(◡𝐹‘𝑋)) = 𝑋)
33	32	ad4antr 732	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) → (𝐹‘(◡𝐹‘𝑋)) = 𝑋)
34	14	simp2d 1143	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(◡𝐹‘𝑌)) = 𝑌)
35	34	ad4antr 732	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) → (𝐹‘(◡𝐹‘𝑌)) = 𝑌)
36	33, 35	opeq12d 4835	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) → ⟨(𝐹‘(◡𝐹‘𝑋)), (𝐹‘(◡𝐹‘𝑌))⟩ = ⟨𝑋, 𝑌⟩)
37	18	simp2d 1143	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(◡𝐹‘𝑍)) = 𝑍)
38	37	ad4antr 732	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) → (𝐹‘(◡𝐹‘𝑍)) = 𝑍)
39	36, 38	oveq12d 7374	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) → (⟨(𝐹‘(◡𝐹‘𝑋)), (𝐹‘(◡𝐹‘𝑌))⟩ ∙ (𝐹‘(◡𝐹‘𝑍))) = (⟨𝑋, 𝑌⟩ ∙ 𝑍))
40		simpr 484	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) → (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁)
41		simpllr 775	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) → (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀)
42	39, 40, 41	oveq123d 7377	. . . . 5 ⊢ (((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) → ((((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛)(⟨(𝐹‘(◡𝐹‘𝑋)), (𝐹‘(◡𝐹‘𝑌))⟩ ∙ (𝐹‘(◡𝐹‘𝑍)))(((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚)) = (𝑁(⟨𝑋, 𝑌⟩ ∙ 𝑍)𝑀))
43	31, 42	eqtr2d 2770	. . . 4 ⊢ (((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) → (𝑁(⟨𝑋, 𝑌⟩ ∙ 𝑍)𝑀) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑍))‘(𝑛(⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩(comp‘𝐷)(◡𝐹‘𝑍))𝑚)))
44		relfunc 17784	. . . . . . . 8 ⊢ Rel (𝐷 Func 𝐸)
45	44	brrelex1i 5678	. . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → 𝐹 ∈ V)
46	4, 45	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
47		imasubc.k	. . . . . 6 ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
48	7, 8, 9, 17, 46, 47	imaf1hom 49295	. . . . 5 ⊢ (𝜑 → (𝑋𝐾𝑍) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑍)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑍))))
49	48	ad4antr 732	. . . 4 ⊢ (((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) → (𝑋𝐾𝑍) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑍)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑍))))
50	28, 43, 49	3eltr4d 2849	. . 3 ⊢ (((((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) ∧ 𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))) ∧ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁) → (𝑁(⟨𝑋, 𝑌⟩ ∙ 𝑍)𝑀) ∈ (𝑋𝐾𝑍))
51	1, 2, 24, 4, 15, 19	funcf2 17790	. . . . . 6 ⊢ (𝜑 → ((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍)):((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))⟶((𝐹‘(◡𝐹‘𝑌))(Hom ‘𝐸)(𝐹‘(◡𝐹‘𝑍))))
52	51	ffund 6664	. . . . 5 ⊢ (𝜑 → Fun ((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍)))
53		imaf1co.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (𝑌𝐾𝑍))
54	7, 8, 13, 17, 46, 47	imaf1hom 49295	. . . . . 6 ⊢ (𝜑 → (𝑌𝐾𝑍) = (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍)) “ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))))
55	53, 54	eleqtrd 2836	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍)) “ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))))
56		fvelima 6897	. . . . 5 ⊢ ((Fun ((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍)) ∧ 𝑁 ∈ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍)) “ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍)))) → ∃𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))(((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁)
57	52, 55, 56	syl2anc 584	. . . 4 ⊢ (𝜑 → ∃𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))(((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁)
58	57	ad2antrr 726	. . 3 ⊢ (((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) → ∃𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))(((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁)
59	50, 58	r19.29a 3142	. 2 ⊢ (((𝜑 ∧ 𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))) ∧ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀) → (𝑁(⟨𝑋, 𝑌⟩ ∙ 𝑍)𝑀) ∈ (𝑋𝐾𝑍))
60	1, 2, 24, 4, 11, 15	funcf2 17790	. . . 4 ⊢ (𝜑 → ((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)):((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))⟶((𝐹‘(◡𝐹‘𝑋))(Hom ‘𝐸)(𝐹‘(◡𝐹‘𝑌))))
61	60	ffund 6664	. . 3 ⊢ (𝜑 → Fun ((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)))
62		imaf1co.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐾𝑌))
63	7, 8, 9, 13, 46, 47	imaf1hom 49295	. . . 4 ⊢ (𝜑 → (𝑋𝐾𝑌) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))))
64	62, 63	eleqtrd 2836	. . 3 ⊢ (𝜑 → 𝑀 ∈ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))))
65		fvelima 6897	. . 3 ⊢ ((Fun ((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) ∧ 𝑀 ∈ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌)))) → ∃𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))(((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀)
66	61, 64, 65	syl2anc 584	. 2 ⊢ (𝜑 → ∃𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))(((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀)
67	59, 66	r19.29a 3142	1 ⊢ (𝜑 → (𝑁(⟨𝑋, 𝑌⟩ ∙ 𝑍)𝑀) ∈ (𝑋𝐾𝑍))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3058  Vcvv 3438  {csn 4578  ⟨cop 4584  ∪ ciun 4944   class class class wbr 5096   × cxp 5620  ^◡ccnv 5621   “ cima 5625  Fun wfun 6484  ⟶wf 6486  –1-1→wf1 6487  ‘cfv 6490  (class class class)co 7356   ∈ cmpo 7358  Basecbs 17134  Hom chom 17186  compcco 17187  Catccat 17585   Func cfunc 17776

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 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8763  df-ixp 8834  df-cat 17589  df-func 17780

This theorem is referenced by:  imasubc3  49343