Theorem cnvepres 37680

Description: Restricted converse epsilon relation as a class of ordered pairs. (Contributed by Peter Mazsa, 10-Feb-2018.)

Assertion

Ref	Expression
cnvepres	⊢ (◡ E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem cnvepres

Step	Hyp	Ref	Expression
1		dfres2 6035	. 2 ⊢ (◡ E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥◡ E 𝑦)}
2		brcnvep 37646	. . . . 5 ⊢ (𝑥 ∈ V → (𝑥◡ E 𝑦 ↔ 𝑦 ∈ 𝑥))
3	2	elv 3474	. . . 4 ⊢ (𝑥◡ E 𝑦 ↔ 𝑦 ∈ 𝑥)
4	3	anbi2i 622	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥◡ E 𝑦) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥))
5	4	opabbii 5208	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥◡ E 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
6	1, 5	eqtri 2754	1 ⊢ (◡ E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468   class class class wbr 5141  {copab 5203   E cep 5572  ^◡ccnv 5668   ↾ cres 5671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-eprel 5573  df-xp 5675  df-rel 5676  df-cnv 5677  df-res 5681

This theorem is referenced by:  rncnvepres  37685  n0el2  37715  cnvepresex  37716