Theorem cnvepresex 39270

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 39238	. 2 ⊢ (◡ E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
2		id 23	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
3		abid2 2898	. . . . 5 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑥} = 𝑥
4		vex 3455	. . . . 5 ⊢ 𝑥 ∈ V
5	3, 4	eqeltri 2857	. . . 4 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑥} ∈ V
6	5	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑦 ∣ 𝑦 ∈ 𝑥} ∈ V)
7	2, 6	opabex3d 7978	. 2 ⊢ (𝐴 ∈ 𝑉 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} ∈ V)
8	1, 7	eqeltrid 2865	1 ⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  {cab 2739  Vcvv 3451  {copab 5167   E cep 5550  ^◡ccnv 5650   ↾ cres 5653

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 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-pow 5327  ax-pr 5391  ax-un 7752

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 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3900  df-un 3902  df-in 3904  df-ss 3915  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-eprel 5551  df-xp 5657  df-rel 5658  df-cnv 5659  df-res 5663

This theorem is used by:  cnvepima  39271  xrncnvepresex  39365  1cosscnvepresex  39445  cnvepresdmqss  39671  eleldisjseldisj  39763  mpets2  39889