Theorem dfrel4v 6053

Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6771 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.)

Assertion

Ref	Expression
dfrel4v	⊢ (Rel 𝑅 ↔ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})

Distinct variable group:   𝑥,𝑦,𝑅

Proof of Theorem dfrel4v

Step	Hyp	Ref	Expression
1		dfrel2 6052	. 2 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
2		eqcom 2744	. 2 ⊢ (◡◡𝑅 = 𝑅 ↔ 𝑅 = ◡◡𝑅)
3		cnvcnv3 6051	. . 3 ⊢ ◡◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
4	3	eqeq2i 2750	. 2 ⊢ (𝑅 = ◡◡𝑅 ↔ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
5	1, 2, 4	3bitri 300	1 ⊢ (Rel 𝑅 ↔ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})

Syntax hints:   ↔ wb 209   = wceq 1543   class class class wbr 5053  {copab 5115  ^◡ccnv 5550  Rel wrel 5556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-cnv 5559

This theorem is referenced by:  dfrel4  6054  dffn5  6771  fsplit  7885  fsplitOLD  7886  pwsle  16997  tgphaus  23014  fneer  34279  inxp2  36234  dfxrn2  36243  1cosscnvxrn  36330  dfafn5a  44324  sprsymrelfo  44622