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Theorem relbrcnvg 5967
Description: When 𝑅 is a relation, the sethood assumptions on brcnv 5752 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 5966 . . . 4 Rel 𝑅
21brrelex12i 5606 . . 3 (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
32a1i 11 . 2 (Rel 𝑅 → (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
4 brrelex12 5603 . . . 4 ((Rel 𝑅𝐵𝑅𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
54ancomd 464 . . 3 ((Rel 𝑅𝐵𝑅𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
65ex 415 . 2 (Rel 𝑅 → (𝐵𝑅𝐴 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
7 brcnvg 5749 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵𝐵𝑅𝐴))
87a1i 11 . 2 (Rel 𝑅 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵𝐵𝑅𝐴)))
93, 6, 8pm5.21ndd 383 1 (Rel 𝑅 → (𝐴𝑅𝐵𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wcel 2110  Vcvv 3494   class class class wbr 5065  ccnv 5553  Rel wrel 5559
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-xp 5560  df-rel 5561  df-cnv 5562
This theorem is referenced by:  eliniseg2  5968  relbrcnv  5969  isinv  17029  releleccnv  35517  relcnveq2  35579  elrelscnveq2  35732  eqvrelsym  35839  brco2f1o  40380  brco3f1o  40381  ntrclsnvobr  40400  neicvgel1  40467
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