Theorem cnvepresex 39459

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 39427	. 2 ⊢ (◡ E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
2		id 23	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
3		abid2 2901	. . . . 5 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑥} = 𝑥
4		vex 3458	. . . . 5 ⊢ 𝑥 ∈ V
5	3, 4	eqeltri 2860	. . . 4 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑥} ∈ V
6	5	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑦 ∣ 𝑦 ∈ 𝑥} ∈ V)
7	2, 6	opabex3d 7998	. 2 ⊢ (𝐴 ∈ 𝑉 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} ∈ V)
8	1, 7	eqeltrid 2868	1 ⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147  {cab 2741  Vcvv 3454  {copab 5170   E cep 5554  ^◡ccnv 5654   ↾ cres 5657

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-rep 5235  ax-sep 5252  ax-pow 5331  ax-pr 5395  ax-un 7772

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-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-eprel 5555  df-xp 5661  df-rel 5662  df-cnv 5663  df-res 5667

This theorem is used by:  cnvepima  39460  xrncnvepresex  39554  1cosscnvepresex  39634  cnvepresdmqss  39860  eleldisjseldisj  39952  mpets2  40078