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Theorem cnven 8573
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 485 . 2 ((Rel 𝐴𝐴𝑉) → 𝐴𝑉)
2 cnvexg 7618 . . 3 (𝐴𝑉𝐴 ∈ V)
32adantl 482 . 2 ((Rel 𝐴𝐴𝑉) → 𝐴 ∈ V)
4 cnvf1o 7795 . . 3 (Rel 𝐴 → (𝑥𝐴 {𝑥}):𝐴1-1-onto𝐴)
54adantr 481 . 2 ((Rel 𝐴𝐴𝑉) → (𝑥𝐴 {𝑥}):𝐴1-1-onto𝐴)
6 f1oen2g 8514 . 2 ((𝐴𝑉𝐴 ∈ V ∧ (𝑥𝐴 {𝑥}):𝐴1-1-onto𝐴) → 𝐴𝐴)
71, 3, 5, 6syl3anc 1363 1 ((Rel 𝐴𝐴𝑉) → 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  Vcvv 3492  {csn 4557   cuni 4830   class class class wbr 5057  cmpt 5137  ccnv 5547  Rel wrel 5553  1-1-ontowf1o 6347  cen 8494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-1st 7678  df-2nd 7679  df-en 8498
This theorem is referenced by:  cnvct  8574  cnvfi  8794  lgsquadlem3  25885
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