Theorem relbrcnv 6064

Description: When 𝑅 is a relation, the sethood assumptions on brcnv 5829 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 6062	. 2 ⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))
3	1, 2	ax-mp 5	1 ⊢ (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)

Syntax hints:   ↔ wb 206   class class class wbr 5086  ^◡ccnv 5621  Rel wrel 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5628  df-rel 5629  df-cnv 5630

This theorem is referenced by:  compssiso  10285  fneval  36540  br1cnvinxp  38571  brcnvep  38582  brid  38624  brcnvrabga  38654  br1cnvxrn2  38731  br1cnvssrres  38897  brcnvssr  38898  brco2f1o  44462  brco3f1o  44463  neicvgnvor  44546