Theorem cnvepresex 39071

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 39039	. 2 ⊢ (◡ E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
2		id 23	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
3		abid2 2899	. . . . 5 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑥} = 𝑥
4		vex 3457	. . . . 5 ⊢ 𝑥 ∈ V
5	3, 4	eqeltri 2858	. . . 4 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑥} ∈ V
6	5	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑦 ∣ 𝑦 ∈ 𝑥} ∈ V)
7	2, 6	opabex3d 7965	. 2 ⊢ (𝐴 ∈ 𝑉 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} ∈ V)
8	1, 7	eqeltrid 2866	1 ⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  {cab 2740  Vcvv 3453  {copab 5171   E cep 5558  ^◡ccnv 5658   ↾ cres 5661

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-pow 5334  ax-pr 5402  ax-un 7739

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 2100  df-mo 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-eprel 5559  df-xp 5665  df-rel 5666  df-cnv 5667  df-res 5671

This theorem is used by:  cnvepima  39072  xrncnvepresex  39166  1cosscnvepresex  39246  cnvepresdmqss  39472  eleldisjseldisj  39564  mpets2  39690