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Theorem relbrcnvg 5982
Description: When 𝑅 is a relation, the sethood assumptions on brcnv 5760 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
relbrcnvg (Rel 𝑅 → (𝐴𝑅𝐵𝐵𝑅𝐴))

Proof of Theorem relbrcnvg
StepHypRef Expression
1 relcnv 5981 . . . 4 Rel 𝑅
21brrelex12i 5613 . . 3 (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
32a1i 11 . 2 (Rel 𝑅 → (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
4 brrelex12 5610 . . . 4 ((Rel 𝑅𝐵𝑅𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
54ancomd 465 . . 3 ((Rel 𝑅𝐵𝑅𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
65ex 416 . 2 (Rel 𝑅 → (𝐵𝑅𝐴 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
7 brcnvg 5757 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵𝐵𝑅𝐴))
87a1i 11 . 2 (Rel 𝑅 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵𝐵𝑅𝐴)))
93, 6, 8pm5.21ndd 384 1 (Rel 𝑅 → (𝐴𝑅𝐵𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2111  Vcvv 3415   class class class wbr 5062  ccnv 5559  Rel wrel 5565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709  ax-sep 5201  ax-nul 5208  ax-pr 5331
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3417  df-dif 3878  df-un 3880  df-in 3882  df-ss 3892  df-nul 4247  df-if 4449  df-sn 4551  df-pr 4553  df-op 4557  df-br 5063  df-opab 5125  df-xp 5566  df-rel 5567  df-cnv 5568
This theorem is referenced by:  eliniseg2  5983  relbrcnv  5984  isinv  17278  releleccnv  36146  relcnveq2  36208  elrelscnveq2  36361  eqvrelsym  36468  brco2f1o  41334  brco3f1o  41335  ntrclsnvobr  41354  neicvgel1  41421
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