Theorem dfrel4v 4715

Description: A relation can be expressed as the set of ordered pairs in it. (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 4714	. 2 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
2		eqcom 2039	. 2 ⊢ (◡◡𝑅 = 𝑅 ↔ 𝑅 = ◡◡𝑅)
3		cnvcnv3 4713	. . 3 ⊢ ◡◡𝑅 = {⟨x, y⟩ ∣ x𝑅y}
4	3	eqeq2i 2047	. 2 ⊢ (𝑅 = ◡◡𝑅 ↔ 𝑅 = {⟨x, y⟩ ∣ x𝑅y})
5	1, 2, 4	3bitri 195	1 ⊢ (Rel 𝑅 ↔ 𝑅 = {⟨x, y⟩ ∣ x𝑅y})

Syntax hints:   ↔ wb 98   = wceq 1242   class class class wbr 3755  {copab 3808  ^◡ccnv 4287  Rel wrel 4293

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296

This theorem is referenced by:  dffn5im  5162