Theorem parhomid 50957

Description: The identity value belongs to each endomorphism hom-set. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 16-Sep-2026.)

Hypotheses

Ref	Expression
parhomval.h	⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
parhomid.j	⊢ 𝐽 ∈ V
parhomid.u	⊢ (𝜑 → 𝑈 ∈ 𝐵)

Assertion

Ref	Expression
parhomid	⊢ (𝜑 → 𝐽 ∈ (𝑈𝐻𝑈))

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

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

Proof of Theorem parhomid

Step	Hyp	Ref	Expression
1		parhomid.j	. . 3 ⊢ 𝐽 ∈ V
2	1	snid 4623	. 2 ⊢ 𝐽 ∈ {𝐽}
3		parhomid.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐵)
4		parhomval.h	. . . 4 ⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
5	4	parhomidval 50951	. . 3 ⊢ (𝑈 ∈ 𝐵 → (𝑈𝐻𝑈) = {𝐽})
6	3, 5	syl 18	. 2 ⊢ (𝜑 → (𝑈𝐻𝑈) = {𝐽})
7	2, 6	eleqtrrid 2871	1 ⊢ (𝜑 → 𝐽 ∈ (𝑈𝐻𝑈))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ∅c0 4281  ifcif 4485  {csn 4588  (class class class)co 7443   ∈ cmpo 7445

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6512  df-fun 6560  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448

This theorem is used by:  parcatid2  50968  parcatid  50969  parmapfunc2  50979  parmapfunc  50980