Theorem parmapcl 50862

Description: A map fixing the identity value and preserving the parallel-arrow class preserves each hom-set. (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, ∅)))
parmapcl.f	⊢ (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))
parmapcl.i	⊢ (𝜑 → (𝐴‘𝐽) = 𝐽)
parmapcl.a	⊢ ((𝜑 ∧ 𝐹 ∈ 𝑆) → (𝐴‘𝐹) ∈ 𝑆)

Assertion

Ref	Expression
parmapcl	⊢ (𝜑 → (𝐴‘𝐹) ∈ (𝑈𝐻𝑉))

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

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

Proof of Theorem parmapcl

Step	Hyp	Ref	Expression
1		parmapcl.i	. . . . 5 ⊢ (𝜑 → (𝐴‘𝐽) = 𝐽)
2	1	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐴‘𝐽) = 𝐽)
3		parmapcl.h	. . . . . 6 ⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
4		parmapcl.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))
5	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝐹 ∈ (𝑈𝐻𝑉))
6		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝑈 = 𝑉)
7	3, 5, 6	parhomideq 50841	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝐹 = 𝐽)
8	7	fveq2d 6889	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐴‘𝐹) = (𝐴‘𝐽))
9	2, 8, 7	3eqtr4d 2805	. . 3 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐴‘𝐹) = 𝐹)
10	9, 5	eqeltrd 2860	. 2 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐴‘𝐹) ∈ (𝑈𝐻𝑉))
11	4	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → 𝐹 ∈ (𝑈𝐻𝑉))
12	3	parhomvaln0 50844	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑈𝐻𝑉) → (𝑈𝐻𝑉) = if(𝑈 = 𝑉, {𝐽}, if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅)))
13	11, 12	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → (𝑈𝐻𝑉) = if(𝑈 = 𝑉, {𝐽}, if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅)))
14		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → ¬ 𝑈 = 𝑉)
15	14	iffalsed 4492	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → if(𝑈 = 𝑉, {𝐽}, if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅)) = if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅))
16	13, 15	eqtrd 2795	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → (𝑈𝐻𝑉) = if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅))
17	11, 16	eleqtrd 2862	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → 𝐹 ∈ if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅))
18		elif0 50013	. . . . . 6 ⊢ (𝐹 ∈ if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅) ↔ ((𝑈 = 𝑋 ∧ 𝑉 = 𝑌) ∧ 𝐹 ∈ 𝑆))
19	17, 18	sylib 221	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → ((𝑈 = 𝑋 ∧ 𝑉 = 𝑌) ∧ 𝐹 ∈ 𝑆))
20	19	simprd 501	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → 𝐹 ∈ 𝑆)
21		parmapcl.a	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝑆) → (𝐴‘𝐹) ∈ 𝑆)
22	20, 21	syldan 603	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → (𝐴‘𝐹) ∈ 𝑆)
23	19	simpld 500	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → (𝑈 = 𝑋 ∧ 𝑉 = 𝑌))
24	23	iftrued 4489	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅) = 𝑆)
25	16, 24	eqtrd 2795	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → (𝑈𝐻𝑉) = 𝑆)
26	22, 25	eleqtrrd 2863	. 2 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → (𝐴‘𝐹) ∈ (𝑈𝐻𝑉))
27	10, 26	pm2.61dan 825	1 ⊢ (𝜑 → (𝐴‘𝐹) ∈ (𝑈𝐻𝑉))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∅c0 4278  ifcif 4481  {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-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-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425

This theorem is used by:  parmapco2  50863  parmapco  50864  parmapfunc2  50865  parmapfunc  50866