Theorem relbrcnv 6112

Description: When 𝑅 is a relation, the sethood assumptions on brcnv 5885 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)

Hypothesis

Ref	Expression
relbrcnv.1	⊢ Rel 𝑅

Assertion

Ref	Expression
relbrcnv	⊢ (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)

Proof of Theorem relbrcnv

Step	Hyp	Ref	Expression
1		relbrcnv.1	. 2 ⊢ Rel 𝑅
2		relbrcnvg 6110	. 2 ⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))
3	1, 2	ax-mp 5	1 ⊢ (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)

Syntax hints:   ↔ wb 205   class class class wbr 5149  ^◡ccnv 5677  Rel wrel 5683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685  df-cnv 5686

This theorem is referenced by:  compssiso  10399  fneval  35964  br1cnvinxp  37855  brcnvep  37864  brid  37905  brcnvrabga  37941  br1cnvxrn2  37995  br1cnvssrres  38104  brcnvssr  38105  brco2f1o  43601  brco3f1o  43602  neicvgnvor  43685