Theorem cnvepres 38043

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 6055	. 2 ⊢ (◡ E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥◡ E 𝑦)}
2		brcnvep 38010	. . . . 5 ⊢ (𝑥 ∈ V → (𝑥◡ E 𝑦 ↔ 𝑦 ∈ 𝑥))
3	2	elv 3478	. . . 4 ⊢ (𝑥◡ E 𝑦 ↔ 𝑦 ∈ 𝑥)
4	3	anbi2i 621	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥◡ E 𝑦) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥))
5	4	opabbii 5223	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥◡ E 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
6	1, 5	eqtri 2757	1 ⊢ (◡ E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2100  Vcvv 3472   class class class wbr 5156  {copab 5218   E cep 5589  ^◡ccnv 5685   ↾ cres 5688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-ext 2700  ax-sep 5307  ax-nul 5314  ax-pr 5437

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2062  df-clab 2707  df-cleq 2721  df-clel 2806  df-ne 2934  df-ral 3055  df-rex 3064  df-rab 3429  df-v 3474  df-dif 3962  df-un 3964  df-in 3966  df-ss 3976  df-nul 4336  df-if 4537  df-sn 4637  df-pr 4639  df-op 4643  df-br 5157  df-opab 5219  df-eprel 5590  df-xp 5692  df-rel 5693  df-cnv 5694  df-res 5698

This theorem is referenced by:  rncnvepres  38048  n0el2  38078  cnvepresex  38079