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

Theorem cnven 7979
Description: A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)
Assertion
Ref Expression
cnven ((Rel 𝐴𝐴𝑉) → 𝐴𝐴)

Proof of Theorem cnven
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpr 477 . 2 ((Rel 𝐴𝐴𝑉) → 𝐴𝑉)
2 cnvexg 7062 . . 3 (𝐴𝑉𝐴 ∈ V)
32adantl 482 . 2 ((Rel 𝐴𝐴𝑉) → 𝐴 ∈ V)
4 cnvf1o 7224 . . 3 (Rel 𝐴 → (𝑥𝐴 {𝑥}):𝐴1-1-onto𝐴)
54adantr 481 . 2 ((Rel 𝐴𝐴𝑉) → (𝑥𝐴 {𝑥}):𝐴1-1-onto𝐴)
6 f1oen2g 7919 . 2 ((𝐴𝑉𝐴 ∈ V ∧ (𝑥𝐴 {𝑥}):𝐴1-1-onto𝐴) → 𝐴𝐴)
71, 3, 5, 6syl3anc 1323 1 ((Rel 𝐴𝐴𝑉) → 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  Vcvv 3186  {csn 4150   cuni 4404   class class class wbr 4615  cmpt 4675  ccnv 5075  Rel wrel 5081  1-1-ontowf1o 5848  cen 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-1st 7116  df-2nd 7117  df-en 7903
This theorem is referenced by:  cnvct  7980  cnvfi  8195  lgsquadlem3  25014
  Copyright terms: Public domain W3C validator