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Theorem relbrcnvg 5745
 Description: When 𝑅 is a relation, the sethood assumptions on brcnv 5537 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 5744 . . . 4 Rel 𝑅
21brrelex12i 5392 . . 3 (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
32a1i 11 . 2 (Rel 𝑅 → (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
4 brrelex12 5389 . . . 4 ((Rel 𝑅𝐵𝑅𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
54ancomd 455 . . 3 ((Rel 𝑅𝐵𝑅𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
65ex 403 . 2 (Rel 𝑅 → (𝐵𝑅𝐴 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
7 brcnvg 5534 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵𝐵𝑅𝐴))
87a1i 11 . 2 (Rel 𝑅 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵𝐵𝑅𝐴)))
93, 6, 8pm5.21ndd 371 1 (Rel 𝑅 → (𝐴𝑅𝐵𝐵𝑅𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∈ wcel 2166  Vcvv 3414   class class class wbr 4873  ◡ccnv 5341  Rel wrel 5347 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-xp 5348  df-rel 5349  df-cnv 5350 This theorem is referenced by:  eliniseg2  5746  relbrcnv  5747  isinv  16772  releleccnv  34575  relcnveq2  34642  elrelscnveq2  34791  eqvrelsym  34895  brco2f1o  39170  brco3f1o  39171  ntrclsnvobr  39190  neicvgel1  39257
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