Theorem dfrel4v 6137

Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6880 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 6136	. 2 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
2		eqcom 2738	. 2 ⊢ (◡◡𝑅 = 𝑅 ↔ 𝑅 = ◡◡𝑅)
3		cnvcnv3 6135	. . 3 ⊢ ◡◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
4	3	eqeq2i 2744	. 2 ⊢ (𝑅 = ◡◡𝑅 ↔ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
5	1, 2, 4	3bitri 297	1 ⊢ (Rel 𝑅 ↔ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})

Syntax hints:   ↔ wb 206   = wceq 1541   class class class wbr 5089  {copab 5151  ^◡ccnv 5613  Rel wrel 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622

This theorem is referenced by:  dfrel4  6138  dffn5  6880  fsplit  8047  pwsle  17396  tgphaus  24032  fneer  36397  inxp2  38398  dfxrn2  38408  1cosscnvxrn  38576  dfafn5a  47259  sprsymrelfo  47596