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Theorem cnven 8380
 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 7442 . . 3 (𝐴𝑉𝐴 ∈ V)
32adantl 474 . 2 ((Rel 𝐴𝐴𝑉) → 𝐴 ∈ V)
4 cnvf1o 7612 . . 3 (Rel 𝐴 → (𝑥𝐴 {𝑥}):𝐴1-1-onto𝐴)
54adantr 473 . 2 ((Rel 𝐴𝐴𝑉) → (𝑥𝐴 {𝑥}):𝐴1-1-onto𝐴)
6 f1oen2g 8321 . 2 ((𝐴𝑉𝐴 ∈ V ∧ (𝑥𝐴 {𝑥}):𝐴1-1-onto𝐴) → 𝐴𝐴)
71, 3, 5, 6syl3anc 1351 1 ((Rel 𝐴𝐴𝑉) → 𝐴𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   ∈ wcel 2050  Vcvv 3409  {csn 4435  ∪ cuni 4708   class class class wbr 4925   ↦ cmpt 5004  ◡ccnv 5402  Rel wrel 5408  –1-1-onto→wf1o 6184   ≈ cen 8301 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-1st 7499  df-2nd 7500  df-en 8305 This theorem is referenced by:  cnvct  8381  cnvfi  8599  lgsquadlem3  25672
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