Theorem cnvepres 38254

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 6070	. 2 ⊢ (◡ E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥◡ E 𝑦)}
2		brcnvep 38221	. . . . 5 ⊢ (𝑥 ∈ V → (𝑥◡ E 𝑦 ↔ 𝑦 ∈ 𝑥))
3	2	elv 3493	. . . 4 ⊢ (𝑥◡ E 𝑦 ↔ 𝑦 ∈ 𝑥)
4	3	anbi2i 622	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥◡ E 𝑦) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥))
5	4	opabbii 5233	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥◡ E 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
6	1, 5	eqtri 2768	1 ⊢ (◡ E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   class class class wbr 5166  {copab 5228   E cep 5598  ^◡ccnv 5699   ↾ cres 5702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-eprel 5599  df-xp 5706  df-rel 5707  df-cnv 5708  df-res 5712

This theorem is referenced by:  rncnvepres  38259  n0el2  38289  cnvepresex  38290