Theorem parmapfunc2 50865

Description: Variant of parmapfunc 50866. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.) (New usage is discouraged.) (Proof modification is discouraged.)

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
parmapfunc2.i	⊢ 𝐼 = (Id‘𝐶)
parmapfunc2.o	⊢ · = (comp‘𝐶)
parmapfunc2.s	⊢ 𝑆 ∈ V
parmapfunc2.j	⊢ 𝐽 ∈ V
parmapfunc2.p	⊢ 𝑆 ⊆ 𝒫 𝐽
parmapfunc2.n	⊢ 𝑋 ≠ 𝑌
parmapfunc2.k	⊢ 𝐾 = (𝑆 ∪ {𝐽})
parmapfunc2.mv	⊢ 𝑀 = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐾 ↦ (𝑓 ∩ 𝑔))
parmapfunc2.ov	⊢ · = (𝑝 ∈ (𝐵 × 𝐵), 𝑞 ∈ 𝐵 ↦ 𝑀)
parmapfunc2.iv	⊢ 𝐼 = (𝑡 ∈ 𝐵 ↦ 𝐽)

Assertion

Ref	Expression
parmapfunc2	⊢ ( I ↾ 𝐵)(𝐶 Func 𝐶)𝐿

Distinct variable groups:   𝐵,𝑟,𝑠,𝑓,𝑔,𝑝,𝑞,𝑢,𝑣,𝑎,𝑡   𝐽,𝑟,𝑠,𝑓,𝑔,𝑝,𝑞,𝑢,𝑣,𝑎,𝑡   𝑆,𝑟,𝑠,𝑓,𝑔,𝑝,𝑞,𝑢,𝑣,𝑎,𝑡   𝑋,𝑟,𝑠,𝑓,𝑔,𝑝,𝑞,𝑢,𝑣,𝑎,𝑡   𝑌,𝑟,𝑠,𝑓,𝑔,𝑝,𝑞,𝑢,𝑣,𝑎,𝑡   𝐾,𝑟,𝑠,𝑓,𝑔,𝑝,𝑞,𝑢,𝑣,𝑎,𝑡   𝐴,𝑟,𝑠,𝑓,𝑔,𝑝,𝑞,𝑢,𝑣,𝑎,𝑡   𝑓,𝐻,𝑔,𝑝,𝑞,𝑢,𝑣,𝑎,𝑡   𝑀,𝑟,𝑠,𝑝,𝑞,𝑢,𝑣,𝑎,𝑡   𝑢, · ,𝑣,𝑎,𝑡   𝑡,𝐿

Allowed substitution hints:   𝐶(𝑣, 𝑢, 𝑡, 𝑓, 𝑔, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎)   · (𝑓, 𝑔, 𝑠, 𝑟, 𝑞, 𝑝)   𝐻(𝑠, 𝑟)   𝐼(𝑣, 𝑢, 𝑡, 𝑓, 𝑔, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎)   𝐿(𝑣, 𝑢, 𝑓, 𝑔, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎)   𝑀(𝑓, 𝑔)

Proof of Theorem parmapfunc2

Dummy variables 𝑥 𝑦 𝑧 𝑖 𝑗 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		parmapfunc2.i	. . 3 ⊢ 𝐼 = (Id‘𝐶)
4		parmapfunc2.o	. . 3 ⊢ · = (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		parmapfunc2.iv	. . . . . . 7 ⊢ 𝐼 = (𝑡 ∈ 𝐵 ↦ 𝐽)
35		eqidd 2761	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ 𝐵) ∧ 𝑡 = 𝑥) → 𝐽 = 𝐽)
36		simpr 490	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
37		parmapfunc2.j	. . . . . . . 8 ⊢ 𝐽 ∈ V
38	37	a1i 11	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → 𝐽 ∈ V)
39	34, 35, 36, 38	fvmptd2 7002	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑥) = 𝐽)
40	39	fveq2d 6889	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ((𝑥𝐿𝑥)‘(𝐼‘𝑥)) = ((𝑥𝐿𝑥)‘𝐽))
41	16, 37, 36	parhomid 50843	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → 𝐽 ∈ (𝑥𝐻𝑥))
42	10, 36, 36, 41	fvmpomptd 50074	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ((𝑥𝐿𝑥)‘𝐽) = (𝐴‘𝐽))
43	40, 42	eqtrd 2795	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ((𝑥𝐿𝑥)‘(𝐼‘𝑥)) = (𝐴‘𝐽))
44	27	adantl 487	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → (( I ↾ 𝐵)‘𝑥) = 𝑥)
45	44	fveq2d 6889	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → (𝐼‘(( I ↾ 𝐵)‘𝑥)) = (𝐼‘𝑥))
46	45, 39	eqtrd 2795	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → (𝐼‘(( I ↾ 𝐵)‘𝑥)) = 𝐽)
47	18, 43, 46	3eqtr4a 2821	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ((𝑥𝐿𝑥)‘(𝐼‘𝑥)) = (𝐼‘(( I ↾ 𝐵)‘𝑥)))
48		parmapfunc2.mv	. . . . 5 ⊢ 𝑀 = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐾 ↦ (𝑓 ∩ 𝑔))
49		parmapfunc2.ov	. . . . 5 ⊢ · = (𝑝 ∈ (𝐵 × 𝐵), 𝑞 ∈ 𝐵 ↦ 𝑀)
50		parmapfunc2.k	. . . . . . . 8 ⊢ 𝐾 = (𝑆 ∪ {𝐽})
51		parmapfunc2.s	. . . . . . . . 9 ⊢ 𝑆 ∈ V
52		snex 5396	. . . . . . . . 9 ⊢ {𝐽} ∈ V
53	51, 52	unex 7761	. . . . . . . 8 ⊢ (𝑆 ∪ {𝐽}) ∈ V
54	50, 53	eqeltri 2856	. . . . . . 7 ⊢ 𝐾 ∈ V
55	54, 54	mpoex 8092	. . . . . 6 ⊢ (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐾 ↦ (𝑓 ∩ 𝑔)) ∈ V
56	48, 55	eqeltri 2856	. . . . 5 ⊢ 𝑀 ∈ V
57		simp21 1225	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → 𝑥 ∈ 𝐵)
58		simp22 1226	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → 𝑦 ∈ 𝐵)
59		simp23 1227	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → 𝑧 ∈ 𝐵)
60		simp3l 1220	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → 𝑖 ∈ (𝑥𝐻𝑦))
61	18	a1i 11	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → (𝐴‘𝐽) = 𝐽)
62		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑡 = 𝑖 → (𝐴‘𝑡) = (𝐴‘𝑖))
63	62	eleq1d 2845	. . . . . . . . 9 ⊢ (𝑡 = 𝑖 → ((𝐴‘𝑡) ∈ 𝑆 ↔ (𝐴‘𝑖) ∈ 𝑆))
64	63, 22	vtoclri 3544	. . . . . . . 8 ⊢ (𝑖 ∈ 𝑆 → (𝐴‘𝑖) ∈ 𝑆)
65	64	adantl 487	. . . . . . 7 ⊢ (((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) ∧ 𝑖 ∈ 𝑆) → (𝐴‘𝑖) ∈ 𝑆)
66	16, 60, 61, 65	parmapcl 50862	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → (𝐴‘𝑖) ∈ (𝑥𝐻𝑦))
67	16, 50, 66	parhomcl 50839	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → (𝐴‘𝑖) ∈ 𝐾)
68		simp3r 1221	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → 𝑗 ∈ (𝑦𝐻𝑧))
69		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑡 = 𝑗 → (𝐴‘𝑡) = (𝐴‘𝑗))
70	69	eleq1d 2845	. . . . . . . . 9 ⊢ (𝑡 = 𝑗 → ((𝐴‘𝑡) ∈ 𝑆 ↔ (𝐴‘𝑗) ∈ 𝑆))
71	70, 22	vtoclri 3544	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑆 → (𝐴‘𝑗) ∈ 𝑆)
72	71	adantl 487	. . . . . . 7 ⊢ (((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) ∧ 𝑗 ∈ 𝑆) → (𝐴‘𝑗) ∈ 𝑆)
73	16, 68, 61, 72	parmapcl 50862	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → (𝐴‘𝑗) ∈ (𝑦𝐻𝑧))
74	16, 50, 73	parhomcl 50839	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → (𝐴‘𝑗) ∈ 𝐾)
75	48, 49, 56, 57, 58, 59, 67, 74	parcoval 50853	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → ((𝐴‘𝑗)(⟨𝑥, 𝑦⟩ · 𝑧)(𝐴‘𝑖)) = ((𝐴‘𝑗) ∩ (𝐴‘𝑖)))
76	57, 27	syl 18	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → (( I ↾ 𝐵)‘𝑥) = 𝑥)
77	58, 29	syl 18	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
78	76, 77	opeq12d 4840	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → ⟨(( I ↾ 𝐵)‘𝑥), (( I ↾ 𝐵)‘𝑦)⟩ = ⟨𝑥, 𝑦⟩)
79		fvresi 7178	. . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑧) = 𝑧)
80	59, 79	syl 18	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → (( I ↾ 𝐵)‘𝑧) = 𝑧)
81	78, 80	oveq12d 7438	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → (⟨(( I ↾ 𝐵)‘𝑥), (( I ↾ 𝐵)‘𝑦)⟩ · (( I ↾ 𝐵)‘𝑧)) = (⟨𝑥, 𝑦⟩ · 𝑧))
82	10, 58, 59, 68	fvmpomptd 50074	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → ((𝑦𝐿𝑧)‘𝑗) = (𝐴‘𝑗))
83	10, 57, 58, 60	fvmpomptd 50074	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → ((𝑥𝐿𝑦)‘𝑖) = (𝐴‘𝑖))
84	81, 82, 83	oveq123d 7441	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → (((𝑦𝐿𝑧)‘𝑗)(⟨(( I ↾ 𝐵)‘𝑥), (( I ↾ 𝐵)‘𝑦)⟩ · (( I ↾ 𝐵)‘𝑧))((𝑥𝐿𝑦)‘𝑖)) = ((𝐴‘𝑗)(⟨𝑥, 𝑦⟩ · 𝑧)(𝐴‘𝑖)))
85	16, 50, 60	parhomcl 50839	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → 𝑖 ∈ 𝐾)
86	16, 50, 68	parhomcl 50839	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → 𝑗 ∈ 𝐾)
87	48, 49, 56, 57, 58, 59, 85, 86	parcoval 50853	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → (𝑗(⟨𝑥, 𝑦⟩ · 𝑧)𝑖) = (𝑗 ∩ 𝑖))
88	87	fveq2d 6889	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → ((𝑥𝐿𝑧)‘(𝑗(⟨𝑥, 𝑦⟩ · 𝑧)𝑖)) = ((𝑥𝐿𝑧)‘(𝑗 ∩ 𝑖)))
89		parmapfunc2.p	. . . . . . . 8 ⊢ 𝑆 ⊆ 𝒫 𝐽
90	16, 60, 68, 89	parhomco 50852	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → (𝑗 ∩ 𝑖) ∈ (𝑥𝐻𝑧))
91	10, 57, 59, 90	fvmpomptd 50074	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → ((𝑥𝐿𝑧)‘(𝑗 ∩ 𝑖)) = (𝐴‘(𝑗 ∩ 𝑖)))
92	88, 91	eqtrd 2795	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → ((𝑥𝐿𝑧)‘(𝑗(⟨𝑥, 𝑦⟩ · 𝑧)𝑖)) = (𝐴‘(𝑗 ∩ 𝑖)))
93	16, 89, 60, 68, 61, 65, 72	parmapco 50864	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → (𝐴‘(𝑗 ∩ 𝑖)) = ((𝐴‘𝑗) ∩ (𝐴‘𝑖)))
94	92, 93	eqtrd 2795	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → ((𝑥𝐿𝑧)‘(𝑗(⟨𝑥, 𝑦⟩ · 𝑧)𝑖)) = ((𝐴‘𝑗) ∩ (𝐴‘𝑖)))
95	75, 84, 94	3eqtr4rd 2806	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑖 ∈ (𝑥𝐻𝑦) ∧ 𝑗 ∈ (𝑦𝐻𝑧))) → ((𝑥𝐿𝑧)‘(𝑗(⟨𝑥, 𝑦⟩ · 𝑧)𝑖)) = (((𝑦𝐿𝑧)‘𝑗)(⟨(( I ↾ 𝐵)‘𝑥), (( I ↾ 𝐵)‘𝑦)⟩ · (( I ↾ 𝐵)‘𝑧))((𝑥𝐿𝑦)‘𝑖)))
96	1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 9, 12, 33, 47, 95	isfuncd 18042	. 2 ⊢ (⊤ → ( I ↾ 𝐵)(𝐶 Func 𝐶)𝐿)
97	96	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   ≠ wne 2955  ∀wral 3076  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089   ∪ cun 4149  ∅c0 4278  ifcif 4481  𝒫 cpw 4556  {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-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-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-map 8851  df-ixp 8928  df-func 18035

This theorem is used by: (None)