Theorem relbrcnv 6058

Description: When 𝑅 is a relation, the sethood assumptions on brcnv 5825 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 6056	. 2 ⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))
3	1, 2	ax-mp 5	1 ⊢ (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)

Syntax hints:   ↔ wb 206   class class class wbr 5092  ^◡ccnv 5618  Rel wrel 5624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-xp 5625  df-rel 5626  df-cnv 5627

This theorem is referenced by:  compssiso  10268  fneval  36346  br1cnvinxp  38251  brcnvep  38260  brid  38300  brcnvrabga  38330  br1cnvxrn2  38388  br1cnvssrres  38502  brcnvssr  38503  brco2f1o  44025  brco3f1o  44026  neicvgnvor  44109