MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  relbrcnvg Structured version   Visualization version   GIF version

Theorem relbrcnvg 5654
Description: When 𝑅 is a relation, the sethood assumptions on brcnv 5452 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 5653 . . . 4 Rel 𝑅
2 brrelex12 5304 . . . 4 ((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
31, 2mpan 708 . . 3 (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
43a1i 11 . 2 (Rel 𝑅 → (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
5 brrelex12 5304 . . . 4 ((Rel 𝑅𝐵𝑅𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
65ancomd 466 . . 3 ((Rel 𝑅𝐵𝑅𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
76ex 449 . 2 (Rel 𝑅 → (𝐵𝑅𝐴 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
8 brcnvg 5450 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵𝐵𝑅𝐴))
98a1i 11 . 2 (Rel 𝑅 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵𝐵𝑅𝐴)))
104, 7, 9pm5.21ndd 368 1 (Rel 𝑅 → (𝐴𝑅𝐵𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2131  Vcvv 3332   class class class wbr 4796  ccnv 5257  Rel wrel 5263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-br 4797  df-opab 4857  df-xp 5264  df-rel 5265  df-cnv 5266
This theorem is referenced by:  eliniseg2  5655  relbrcnv  5656  isinv  16613  releleccnv  34337  relcnveq2  34410  elrelscnveq2  34558  brco2f1o  38824  brco3f1o  38825  ntrclsnvobr  38844  neicvgel1  38911
  Copyright terms: Public domain W3C validator