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Theorem relbrcnvg 6126
Description: When 𝑅 is a relation, the sethood assumptions on brcnv 5896 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 6125 . . . 4 Rel 𝑅
21brrelex12i 5744 . . 3 (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
32a1i 11 . 2 (Rel 𝑅 → (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
4 brrelex12 5741 . . . 4 ((Rel 𝑅𝐵𝑅𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
54ancomd 461 . . 3 ((Rel 𝑅𝐵𝑅𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
65ex 412 . 2 (Rel 𝑅 → (𝐵𝑅𝐴 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
7 brcnvg 5893 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵𝐵𝑅𝐴))
87a1i 11 . 2 (Rel 𝑅 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵𝐵𝑅𝐴)))
93, 6, 8pm5.21ndd 379 1 (Rel 𝑅 → (𝐴𝑅𝐵𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2106  Vcvv 3478   class class class wbr 5148  ccnv 5688  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697
This theorem is referenced by:  eliniseg2  6127  relbrcnv  6128  isinv  17808  releleccnv  38239  relcnveq2  38305  elrelscnveq2  38475  eqvrelsym  38587  brco2f1o  44022  brco3f1o  44023  ntrclsnvobr  44042  neicvgel1  44109
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