Theorem cnvepresex 38654

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 38622	. 2 ⊢ (◡ E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
2		id 22	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
3		abid2 2874	. . . . 5 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑥} = 𝑥
4		vex 3434	. . . . 5 ⊢ 𝑥 ∈ V
5	3, 4	eqeltri 2833	. . . 4 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑥} ∈ V
6	5	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑦 ∣ 𝑦 ∈ 𝑥} ∈ V)
7	2, 6	opabex3d 7915	. 2 ⊢ (𝐴 ∈ 𝑉 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} ∈ V)
8	1, 7	eqeltrid 2841	1 ⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  {cab 2715  Vcvv 3430  {copab 5148   E cep 5527  ^◡ccnv 5627   ↾ cres 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-pow 5306  ax-pr 5374  ax-un 7686

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-eprel 5528  df-xp 5634  df-rel 5635  df-cnv 5636  df-res 5640

This theorem is referenced by:  cnvepima  38655  xrncnvepresex  38749  1cosscnvepresex  38829  cnvepresdmqss  39055  eleldisjseldisj  39147  mpets2  39273