Theorem parmapco 50864

Description: An identity-fixing map preserving parallel arrows preserves every typed composition. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)

Hypotheses

Ref	Expression
parmapcl.h	⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
parmapco.p	⊢ 𝑆 ⊆ 𝒫 𝐽
parmapco.f	⊢ (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))
parmapco.g	⊢ (𝜑 → 𝐺 ∈ (𝑉𝐻𝑊))
parmapco.i	⊢ (𝜑 → (𝐴‘𝐽) = 𝐽)
parmapco.a	⊢ ((𝜑 ∧ 𝐹 ∈ 𝑆) → (𝐴‘𝐹) ∈ 𝑆)
parmapco.b	⊢ ((𝜑 ∧ 𝐺 ∈ 𝑆) → (𝐴‘𝐺) ∈ 𝑆)

Assertion

Ref	Expression
parmapco	⊢ (𝜑 → (𝐴‘(𝐺 ∩ 𝐹)) = ((𝐴‘𝐺) ∩ (𝐴‘𝐹)))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐽,𝑦   𝑥,𝑆,𝑦   𝑥,𝑈,𝑦   𝑥,𝑉,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥,𝑊,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐴(𝑥, 𝑦)   𝐹(𝑥, 𝑦)   𝐺(𝑥, 𝑦)   𝐻(𝑥, 𝑦)

Proof of Theorem parmapco

Step	Hyp	Ref	Expression
1		parmapcl.h	. . . . . 6 ⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
2		parmapco.p	. . . . . 6 ⊢ 𝑆 ⊆ 𝒫 𝐽
3		parmapco.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (𝑉𝐻𝑊))
4		parmapco.i	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐽) = 𝐽)
5		parmapco.b	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝑆) → (𝐴‘𝐺) ∈ 𝑆)
6	1, 3, 4, 5	parmapcl 50862	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝐺) ∈ (𝑉𝐻𝑊))
7	1, 2, 6	parhomss 50848	. . . . 5 ⊢ (𝜑 → (𝐴‘𝐺) ⊆ 𝐽)
8	7	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐴‘𝐺) ⊆ 𝐽)
9		dfss2 4094	. . . 4 ⊢ ((𝐴‘𝐺) ⊆ 𝐽 ↔ ((𝐴‘𝐺) ∩ 𝐽) = (𝐴‘𝐺))
10	8, 9	sylib 221	. . 3 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → ((𝐴‘𝐺) ∩ 𝐽) = (𝐴‘𝐺))
11		parmapco.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))
12	11	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝐹 ∈ (𝑈𝐻𝑉))
13		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝑈 = 𝑉)
14	1, 12, 13	parhomideq 50841	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝐹 = 𝐽)
15	14	fveq2d 6889	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐴‘𝐹) = (𝐴‘𝐽))
16	4	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐴‘𝐽) = 𝐽)
17	15, 16	eqtrd 2795	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐴‘𝐹) = 𝐽)
18	17	ineq2d 4117	. . 3 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → ((𝐴‘𝐺) ∩ (𝐴‘𝐹)) = ((𝐴‘𝐺) ∩ 𝐽))
19	14	ineq2d 4117	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐺 ∩ 𝐹) = (𝐺 ∩ 𝐽))
20	1, 2, 3	parhomss 50848	. . . . . . 7 ⊢ (𝜑 → 𝐺 ⊆ 𝐽)
21	20	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝐺 ⊆ 𝐽)
22		dfss2 4094	. . . . . 6 ⊢ (𝐺 ⊆ 𝐽 ↔ (𝐺 ∩ 𝐽) = 𝐺)
23	21, 22	sylib 221	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐺 ∩ 𝐽) = 𝐺)
24	19, 23	eqtrd 2795	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐺 ∩ 𝐹) = 𝐺)
25	24	fveq2d 6889	. . 3 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐴‘(𝐺 ∩ 𝐹)) = (𝐴‘𝐺))
26	10, 18, 25	3eqtr4rd 2806	. 2 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐴‘(𝐺 ∩ 𝐹)) = ((𝐴‘𝐺) ∩ (𝐴‘𝐹)))
27		parmapco.a	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝑆) → (𝐴‘𝐹) ∈ 𝑆)
28	1, 11, 4, 27	parmapcl 50862	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝐹) ∈ (𝑈𝐻𝑉))
29	1, 2, 28	parhomss 50848	. . . . 5 ⊢ (𝜑 → (𝐴‘𝐹) ⊆ 𝐽)
30	29	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → (𝐴‘𝐹) ⊆ 𝐽)
31		sseqin2 4120	. . . 4 ⊢ ((𝐴‘𝐹) ⊆ 𝐽 ↔ (𝐽 ∩ (𝐴‘𝐹)) = (𝐴‘𝐹))
32	30, 31	sylib 221	. . 3 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → (𝐽 ∩ (𝐴‘𝐹)) = (𝐴‘𝐹))
33	3	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → 𝐺 ∈ (𝑉𝐻𝑊))
34		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → 𝑉 = 𝑊)
35	1, 33, 34	parhomideq 50841	. . . . . 6 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → 𝐺 = 𝐽)
36	35	fveq2d 6889	. . . . 5 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → (𝐴‘𝐺) = (𝐴‘𝐽))
37	4	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → (𝐴‘𝐽) = 𝐽)
38	36, 37	eqtrd 2795	. . . 4 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → (𝐴‘𝐺) = 𝐽)
39	38	ineq1d 4116	. . 3 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → ((𝐴‘𝐺) ∩ (𝐴‘𝐹)) = (𝐽 ∩ (𝐴‘𝐹)))
40	35	ineq1d 4116	. . . . 5 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → (𝐺 ∩ 𝐹) = (𝐽 ∩ 𝐹))
41	1, 2, 11	parhomss 50848	. . . . . . 7 ⊢ (𝜑 → 𝐹 ⊆ 𝐽)
42	41	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → 𝐹 ⊆ 𝐽)
43		sseqin2 4120	. . . . . 6 ⊢ (𝐹 ⊆ 𝐽 ↔ (𝐽 ∩ 𝐹) = 𝐹)
44	42, 43	sylib 221	. . . . 5 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → (𝐽 ∩ 𝐹) = 𝐹)
45	40, 44	eqtrd 2795	. . . 4 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → (𝐺 ∩ 𝐹) = 𝐹)
46	45	fveq2d 6889	. . 3 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → (𝐴‘(𝐺 ∩ 𝐹)) = (𝐴‘𝐹))
47	32, 39, 46	3eqtr4rd 2806	. 2 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → (𝐴‘(𝐺 ∩ 𝐹)) = ((𝐴‘𝐺) ∩ (𝐴‘𝐹)))
48	1, 11, 3	parhomeqor 50850	. 2 ⊢ (𝜑 → (𝑈 = 𝑉 ∨ 𝑉 = 𝑊))
49	26, 47, 48	mpjaodan 973	1 ⊢ (𝜑 → (𝐴‘(𝐺 ∩ 𝐹)) = ((𝐴‘𝐺) ∩ (𝐴‘𝐹)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   ∩ cin 4089  ∅c0 4278  ifcif 4481  𝒫 cpw 4556  {csn 4583  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422

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-sep 5248  ax-nul 5259  ax-pr 5390

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-rab 3413  df-v 3452  df-sbc 3739  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-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425

This theorem is used by:  parmapfunc2  50865