Theorem parmapfunc 50866

Description: A map fixing the identity value and preserving the parallel-arrow class induces an endofunctor fixing every object. Category membership and the hom construction suffice: every composable pair contains an identity. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)

Hypotheses

Ref	Expression
parmapfunc.b	⊢ 𝐵 = (Base‘𝐶)
parmapfunc.h	⊢ 𝐻 = (Hom ‘𝐶)
parmapfunc.c	⊢ 𝐶 ∈ Cat
parmapfunc.hv	⊢ 𝐻 = (𝑟 ∈ 𝐵, 𝑠 ∈ 𝐵 ↦ if(𝑟 = 𝑠, {𝐽}, if((𝑟 = 𝑋 ∧ 𝑠 = 𝑌), 𝑆, ∅)))
parmapfunc.l	⊢ 𝐿 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑎 ∈ (𝑢𝐻𝑣) ↦ (𝐴‘𝑎)))
parmapfunc.ai	⊢ (𝐴‘𝐽) = 𝐽
parmapfunc.as	⊢ ∀𝑡 ∈ 𝑆 (𝐴‘𝑡) ∈ 𝑆

Assertion

Ref	Expression
parmapfunc	⊢ ( I ↾ 𝐵)(𝐶 Func 𝐶)𝐿

Distinct variable groups:   𝐵,𝑟,𝑠,𝑢,𝑣,𝑎,𝑡   𝐽,𝑟,𝑠,𝑢,𝑣,𝑎,𝑡   𝑆,𝑟,𝑠,𝑢,𝑣,𝑎,𝑡   𝑋,𝑟,𝑠,𝑢,𝑣,𝑎,𝑡   𝑌,𝑟,𝑠,𝑢,𝑣,𝑎,𝑡   𝐴,𝑟,𝑠,𝑢,𝑣,𝑎,𝑡   𝑢,𝐻,𝑣,𝑎,𝑡   𝑡,𝐿

Allowed substitution hints:   𝐶(𝑣, 𝑢, 𝑡, 𝑠, 𝑟, 𝑎)   𝐻(𝑠, 𝑟)   𝐿(𝑣, 𝑢, 𝑠, 𝑟, 𝑎)

Proof of Theorem parmapfunc

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

Step	Hyp	Ref	Expression
1		parmapfunc.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		parmapfunc.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
3		eqid 2760	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
4		eqid 2760	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
5		parmapfunc.c	. . . 4 ⊢ 𝐶 ∈ Cat
6	5	a1i 11	. . 3 ⊢ (⊤ → 𝐶 ∈ Cat)
7		f1oi 6863	. . . 4 ⊢ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵
8		f1of 6824	. . . 4 ⊢ (( I ↾ 𝐵):𝐵–1-1-onto→𝐵 → ( I ↾ 𝐵):𝐵⟶𝐵)
9	7, 8	mp1i 14	. . 3 ⊢ (⊤ → ( I ↾ 𝐵):𝐵⟶𝐵)
10		parmapfunc.l	. . . . 5 ⊢ 𝐿 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑎 ∈ (𝑢𝐻𝑣) ↦ (𝐴‘𝑎)))
11	10	fnmpompt 50072	. . . 4 ⊢ 𝐿 Fn (𝐵 × 𝐵)
12	11	a1i 11	. . 3 ⊢ (⊤ → 𝐿 Fn (𝐵 × 𝐵))
13		simprl 783	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
14		simprr 785	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
15	10, 13, 14	ovmpomptd 50073	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐿𝑦) = (𝑎 ∈ (𝑥𝐻𝑦) ↦ (𝐴‘𝑎)))
16		parmapfunc.hv	. . . . . 6 ⊢ 𝐻 = (𝑟 ∈ 𝐵, 𝑠 ∈ 𝐵 ↦ if(𝑟 = 𝑠, {𝐽}, if((𝑟 = 𝑋 ∧ 𝑠 = 𝑌), 𝑆, ∅)))
17		simpr 490	. . . . . 6 ⊢ (((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑎 ∈ (𝑥𝐻𝑦)) → 𝑎 ∈ (𝑥𝐻𝑦))
18		parmapfunc.ai	. . . . . . 7 ⊢ (𝐴‘𝐽) = 𝐽
19	18	a1i 11	. . . . . 6 ⊢ (((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑎 ∈ (𝑥𝐻𝑦)) → (𝐴‘𝐽) = 𝐽)
20		fveq2 6885	. . . . . . . . 9 ⊢ (𝑡 = 𝑎 → (𝐴‘𝑡) = (𝐴‘𝑎))
21	20	eleq1d 2845	. . . . . . . 8 ⊢ (𝑡 = 𝑎 → ((𝐴‘𝑡) ∈ 𝑆 ↔ (𝐴‘𝑎) ∈ 𝑆))
22		parmapfunc.as	. . . . . . . 8 ⊢ ∀𝑡 ∈ 𝑆 (𝐴‘𝑡) ∈ 𝑆
23	21, 22	vtoclri 3544	. . . . . . 7 ⊢ (𝑎 ∈ 𝑆 → (𝐴‘𝑎) ∈ 𝑆)
24	23	adantl 487	. . . . . 6 ⊢ ((((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑎 ∈ (𝑥𝐻𝑦)) ∧ 𝑎 ∈ 𝑆) → (𝐴‘𝑎) ∈ 𝑆)
25	16, 17, 19, 24	parmapcl 50862	. . . . 5 ⊢ (((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑎 ∈ (𝑥𝐻𝑦)) → (𝐴‘𝑎) ∈ (𝑥𝐻𝑦))
26	15, 25	fmpt3d 7116	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐿𝑦):(𝑥𝐻𝑦)⟶(𝑥𝐻𝑦))
27		fvresi 7178	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑥) = 𝑥)
28	13, 27	syl 18	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (( I ↾ 𝐵)‘𝑥) = 𝑥)
29		fvresi 7178	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦)
30	14, 29	syl 18	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
31	28, 30	oveq12d 7438	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((( I ↾ 𝐵)‘𝑥)𝐻(( I ↾ 𝐵)‘𝑦)) = (𝑥𝐻𝑦))
32	31	feq3d 6694	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥𝐿𝑦):(𝑥𝐻𝑦)⟶((( I ↾ 𝐵)‘𝑥)𝐻(( I ↾ 𝐵)‘𝑦)) ↔ (𝑥𝐿𝑦):(𝑥𝐻𝑦)⟶(𝑥𝐻𝑦)))
33	26, 32	mpbird 260	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐿𝑦):(𝑥𝐻𝑦)⟶((( I ↾ 𝐵)‘𝑥)𝐻(( I ↾ 𝐵)‘𝑦)))
34	5	a1i 11	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ Cat)
35		simpr 490	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
36	1, 2, 16, 34, 35	paridval 50856	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) = 𝐽)
37	36	fveq2d 6889	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ((𝑥𝐿𝑥)‘((Id‘𝐶)‘𝑥)) = ((𝑥𝐿𝑥)‘𝐽))
38	18	eqcomi 2769	. . . . . . . 8 ⊢ 𝐽 = (𝐴‘𝐽)
39	38	fvexi 6899	. . . . . . 7 ⊢ 𝐽 ∈ V
40	16, 39, 35	parhomid 50843	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → 𝐽 ∈ (𝑥𝐻𝑥))
41	10, 35, 35, 40	fvmpomptd 50074	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ((𝑥𝐿𝑥)‘𝐽) = (𝐴‘𝐽))
42	37, 41	eqtrd 2795	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ((𝑥𝐿𝑥)‘((Id‘𝐶)‘𝑥)) = (𝐴‘𝐽))
43	27	adantl 487	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → (( I ↾ 𝐵)‘𝑥) = 𝑥)
44	43	fveq2d 6889	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘(( I ↾ 𝐵)‘𝑥)) = ((Id‘𝐶)‘𝑥))
45	44, 36	eqtrd 2795	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘(( I ↾ 𝐵)‘𝑥)) = 𝐽)
46	18, 42, 45	3eqtr4a 2821	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ((𝑥𝐿𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐶)‘(( I ↾ 𝐵)‘𝑥)))
47		fvif 6901	. . . 4 ⊢ (𝐴‘if(𝑥 = 𝑦, 𝑛, 𝑚)) = if(𝑥 = 𝑦, (𝐴‘𝑛), (𝐴‘𝑚))
48	5	a1i 11	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → 𝐶 ∈ Cat)
49		simp3l 1220	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → 𝑚 ∈ (𝑥𝐻𝑦))
50		simp3r 1221	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → 𝑛 ∈ (𝑦𝐻𝑧))
51	1, 2, 16, 48, 4, 49, 50	parcoval2 50857	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → (𝑛(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑚) = if(𝑥 = 𝑦, 𝑛, 𝑚))
52	51	fveq2d 6889	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → ((𝑥𝐿𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑚)) = ((𝑥𝐿𝑧)‘if(𝑥 = 𝑦, 𝑛, 𝑚)))
53		simp21 1225	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → 𝑥 ∈ 𝐵)
54		simp23 1227	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → 𝑧 ∈ 𝐵)
55		simp22 1226	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → 𝑦 ∈ 𝐵)
56	1, 2, 4, 48, 53, 55, 54, 49, 50	catcocl 17861	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → (𝑛(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑚) ∈ (𝑥𝐻𝑧))
57	51, 56	eqeltrrd 2861	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → if(𝑥 = 𝑦, 𝑛, 𝑚) ∈ (𝑥𝐻𝑧))
58	10, 53, 54, 57	fvmpomptd 50074	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → ((𝑥𝐿𝑧)‘if(𝑥 = 𝑦, 𝑛, 𝑚)) = (𝐴‘if(𝑥 = 𝑦, 𝑛, 𝑚)))
59	52, 58	eqtrd 2795	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → ((𝑥𝐿𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑚)) = (𝐴‘if(𝑥 = 𝑦, 𝑛, 𝑚)))
60	53, 27	syl 18	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → (( I ↾ 𝐵)‘𝑥) = 𝑥)
61	55, 29	syl 18	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
62	60, 61	opeq12d 4840	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → ⟨(( I ↾ 𝐵)‘𝑥), (( I ↾ 𝐵)‘𝑦)⟩ = ⟨𝑥, 𝑦⟩)
63		fvresi 7178	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑧) = 𝑧)
64	54, 63	syl 18	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → (( I ↾ 𝐵)‘𝑧) = 𝑧)
65	62, 64	oveq12d 7438	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → (⟨(( I ↾ 𝐵)‘𝑥), (( I ↾ 𝐵)‘𝑦)⟩(comp‘𝐶)(( I ↾ 𝐵)‘𝑧)) = (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧))
66	10, 55, 54, 50	fvmpomptd 50074	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → ((𝑦𝐿𝑧)‘𝑛) = (𝐴‘𝑛))
67	10, 53, 55, 49	fvmpomptd 50074	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → ((𝑥𝐿𝑦)‘𝑚) = (𝐴‘𝑚))
68	65, 66, 67	oveq123d 7441	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → (((𝑦𝐿𝑧)‘𝑛)(⟨(( I ↾ 𝐵)‘𝑥), (( I ↾ 𝐵)‘𝑦)⟩(comp‘𝐶)(( I ↾ 𝐵)‘𝑧))((𝑥𝐿𝑦)‘𝑚)) = ((𝐴‘𝑛)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)(𝐴‘𝑚)))
69	18	a1i 11	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → (𝐴‘𝐽) = 𝐽)
70		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑡 = 𝑚 → (𝐴‘𝑡) = (𝐴‘𝑚))
71	70	eleq1d 2845	. . . . . . . . 9 ⊢ (𝑡 = 𝑚 → ((𝐴‘𝑡) ∈ 𝑆 ↔ (𝐴‘𝑚) ∈ 𝑆))
72	71, 22	vtoclri 3544	. . . . . . . 8 ⊢ (𝑚 ∈ 𝑆 → (𝐴‘𝑚) ∈ 𝑆)
73	72	adantl 487	. . . . . . 7 ⊢ (((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) ∧ 𝑚 ∈ 𝑆) → (𝐴‘𝑚) ∈ 𝑆)
74	16, 49, 69, 73	parmapcl 50862	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → (𝐴‘𝑚) ∈ (𝑥𝐻𝑦))
75		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑡 = 𝑛 → (𝐴‘𝑡) = (𝐴‘𝑛))
76	75	eleq1d 2845	. . . . . . . . 9 ⊢ (𝑡 = 𝑛 → ((𝐴‘𝑡) ∈ 𝑆 ↔ (𝐴‘𝑛) ∈ 𝑆))
77	76, 22	vtoclri 3544	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑆 → (𝐴‘𝑛) ∈ 𝑆)
78	77	adantl 487	. . . . . . 7 ⊢ (((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) ∧ 𝑛 ∈ 𝑆) → (𝐴‘𝑛) ∈ 𝑆)
79	16, 50, 69, 78	parmapcl 50862	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → (𝐴‘𝑛) ∈ (𝑦𝐻𝑧))
80	1, 2, 16, 48, 4, 74, 79	parcoval2 50857	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → ((𝐴‘𝑛)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)(𝐴‘𝑚)) = if(𝑥 = 𝑦, (𝐴‘𝑛), (𝐴‘𝑚)))
81	68, 80	eqtrd 2795	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → (((𝑦𝐿𝑧)‘𝑛)(⟨(( I ↾ 𝐵)‘𝑥), (( I ↾ 𝐵)‘𝑦)⟩(comp‘𝐶)(( I ↾ 𝐵)‘𝑧))((𝑥𝐿𝑦)‘𝑚)) = if(𝑥 = 𝑦, (𝐴‘𝑛), (𝐴‘𝑚)))
82	47, 59, 81	3eqtr4a 2821	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → ((𝑥𝐿𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑚)) = (((𝑦𝐿𝑧)‘𝑛)(⟨(( I ↾ 𝐵)‘𝑥), (( I ↾ 𝐵)‘𝑦)⟩(comp‘𝐶)(( I ↾ 𝐵)‘𝑧))((𝑥𝐿𝑦)‘𝑚)))
83	1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 9, 12, 33, 46, 82	isfuncd 18042	. 2 ⊢ (⊤ → ( I ↾ 𝐵)(𝐶 Func 𝐶)𝐿)
84	83	mptru 1576	1 ⊢ ( I ↾ 𝐵)(𝐶 Func 𝐶)𝐿

This proof depends on syntax axioms:   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ⊤wtru 1570   ∈ wcel 2144  ∀wral 3076  ∅c0 4278  ifcif 4481  {csn 4583  ⟨cop 4589   class class class wbr 5102   ↦ cmpt 5185   I cid 5544   × cxp 5648   ↾ cres 5652   Fn wfn 6533  ⟶wf 6534  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  Basecbs 17389  Hom chom 17441  compcco 17442  Catccat 17840  Idccid 17841   Func cfunc 18031

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-map 8851  df-ixp 8928  df-cat 17844  df-cid 17845  df-func 18035

This theorem is used by:  parswapfunc  50869