Theorem imaf1co 49992

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 2765	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
4		imassc.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
5	4	funcrcl2 49916	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
6	5	ad4antr 745	. . . . . 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 49944	. . . . . . . 8 ⊢ (𝜑 → ({(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}) ∧ (𝐹‘(◡𝐹‘𝑋)) = 𝑋 ∧ (◡𝐹‘𝑋) ∈ 𝐵))
11	10	simp3d 1162	. . . . . . 7 ⊢ (𝜑 → (◡𝐹‘𝑋) ∈ 𝐵)
12	11	ad4antr 745	. . . . . 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 49944	. . . . . . . 8 ⊢ (𝜑 → ({(◡𝐹‘𝑌)} = (◡𝐹 “ {𝑌}) ∧ (𝐹‘(◡𝐹‘𝑌)) = 𝑌 ∧ (◡𝐹‘𝑌) ∈ 𝐵))
15	14	simp3d 1162	. . . . . . 7 ⊢ (𝜑 → (◡𝐹‘𝑌) ∈ 𝐵)
16	15	ad4antr 745	. . . . . 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 49944	. . . . . . . 8 ⊢ (𝜑 → ({(◡𝐹‘𝑍)} = (◡𝐹 “ {𝑍}) ∧ (𝐹‘(◡𝐹‘𝑍)) = 𝑍 ∧ (◡𝐹‘𝑍) ∈ 𝐵))
19	18	simp3d 1162	. . . . . . 7 ⊢ (𝜑 → (◡𝐹‘𝑍) ∈ 𝐵)
20	19	ad4antr 745	. . . . . 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 796	. . . . . 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 781	. . . . . 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 17765	. . . . 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 2765	. . . . . . . 8 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
25	1, 2, 24, 4, 11, 19	funcf2 17949	. . . . . . 7 ⊢ (𝜑 → ((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑍)):((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑍))⟶((𝐹‘(◡𝐹‘𝑋))(Hom ‘𝐸)(𝐹‘(◡𝐹‘𝑍))))
26	25	ad4antr 745	. . . . . 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 7237	. . . . 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 700	. . . 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 745	. . . . . 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 17952	. . . . 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 1161	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(◡𝐹‘𝑋)) = 𝑋)
33	32	ad4antr 745	. . . . . . . 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 1161	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(◡𝐹‘𝑌)) = 𝑌)
35	34	ad4antr 745	. . . . . . . 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 4848	. . . . . . 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 1161	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(◡𝐹‘𝑍)) = 𝑍)
38	37	ad4antr 745	. . . . . . 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 7437	. . . . . 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 490	. . . . . 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 788	. . . . . 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 7440	. . . . 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 2801	. . . 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 17943	. . . . . . . 8 ⊢ Rel (𝐷 Func 𝐸)
45	44	brrelex1i 5719	. . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → 𝐹 ∈ V)
46	4, 45	syl 18	. . . . . 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 49945	. . . . 5 ⊢ (𝜑 → (𝑋𝐾𝑍) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑍)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑍))))
49	48	ad4antr 745	. . . 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 2880	. . 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 17949	. . . . . 6 ⊢ (𝜑 → ((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍)):((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))⟶((𝐹‘(◡𝐹‘𝑌))(Hom ‘𝐸)(𝐹‘(◡𝐹‘𝑍))))
52	51	ffund 6714	. . . . 5 ⊢ (𝜑 → Fun ((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍)))
53		imaf1co.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (𝑌𝐾𝑍))
54	7, 8, 13, 17, 46, 47	imaf1hom 49945	. . . . . 6 ⊢ (𝜑 → (𝑌𝐾𝑍) = (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍)) “ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))))
55	53, 54	eleqtrd 2867	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍)) “ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))))
56		fvelima 6950	. . . . 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 596	. . . 4 ⊢ (𝜑 → ∃𝑛 ∈ ((◡𝐹‘𝑌)𝐻(◡𝐹‘𝑍))(((◡𝐹‘𝑌)𝐺(◡𝐹‘𝑍))‘𝑛) = 𝑁)
58	57	ad2antrr 739	. . 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 3175	. 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 17949	. . . 4 ⊢ (𝜑 → ((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)):((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))⟶((𝐹‘(◡𝐹‘𝑋))(Hom ‘𝐸)(𝐹‘(◡𝐹‘𝑌))))
61	60	ffund 6714	. . 3 ⊢ (𝜑 → Fun ((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)))
62		imaf1co.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐾𝑌))
63	7, 8, 9, 13, 46, 47	imaf1hom 49945	. . . 4 ⊢ (𝜑 → (𝑋𝐾𝑌) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))))
64	62, 63	eleqtrd 2867	. . 3 ⊢ (𝜑 → 𝑀 ∈ (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))))
65		fvelima 6950	. . 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 596	. 2 ⊢ (𝜑 → ∃𝑚 ∈ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))(((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌))‘𝑚) = 𝑀)
67	59, 66	r19.29a 3175	1 ⊢ (𝜑 → (𝑁(⟨𝑋, 𝑌⟩ ∙ 𝑍)𝑀) ∈ (𝑋𝐾𝑍))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃wrex 3091  Vcvv 3457  {csn 4591  ⟨cop 4597  ∪ ciun 4958   class class class wbr 5111   × cxp 5661  ^◡ccnv 5662   “ cima 5666  Fun wfun 6534  ⟶wf 6536  –1-1→wf1 6537  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  Basecbs 17293  Hom chom 17345  compcco 17346  Catccat 17744   Func cfunc 17935

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-map 8832  df-ixp 8902  df-cat 17748  df-func 17939

This theorem is used by:  imasubc3  49993