Theorem cnvepresex 38357

Description: Sethood condition for the restricted converse epsilon relation. (Contributed by Peter Mazsa, 24-Sep-2018.)

Assertion

Ref	Expression
cnvepresex	⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V)

Proof of Theorem cnvepresex

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

Step	Hyp	Ref	Expression
1		cnvepres 38321	. 2 ⊢ (◡ E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
2		id 22	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
3		abid2 2873	. . . . 5 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑥} = 𝑥
4		vex 3468	. . . . 5 ⊢ 𝑥 ∈ V
5	3, 4	eqeltri 2831	. . . 4 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑥} ∈ V
6	5	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑦 ∣ 𝑦 ∈ 𝑥} ∈ V)
7	2, 6	opabex3d 7969	. 2 ⊢ (𝐴 ∈ 𝑉 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} ∈ V)
8	1, 7	eqeltrid 2839	1 ⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  {cab 2714  Vcvv 3464  {copab 5186   E cep 5557  ^◡ccnv 5658   ↾ cres 5661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-eprel 5558  df-xp 5665  df-rel 5666  df-cnv 5667  df-res 5671

This theorem is referenced by:  eccnvepex  38358  cnvepimaex  38359  cnvepima  38360  xrncnvepresex  38431  1cosscnvepresex  38444  cnvepresdmqss  38675  eleldisjseldisj  38752  mpets2  38864