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Theorem f1ocnvb 6063
Description: A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and range interchanged. (Contributed by NM, 8-Dec-2003.)
Assertion
Ref Expression
f1ocnvb (Rel 𝐹 → (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴))

Proof of Theorem f1ocnvb
StepHypRef Expression
1 f1ocnv 6062 . 2 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
2 f1ocnv 6062 . . 3 (𝐹:𝐵1-1-onto𝐴𝐹:𝐴1-1-onto𝐵)
3 dfrel2 5502 . . . 4 (Rel 𝐹𝐹 = 𝐹)
4 f1oeq1 6040 . . . 4 (𝐹 = 𝐹 → (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1-onto𝐵))
53, 4sylbi 206 . . 3 (Rel 𝐹 → (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1-onto𝐵))
62, 5syl5ib 233 . 2 (Rel 𝐹 → (𝐹:𝐵1-1-onto𝐴𝐹:𝐴1-1-onto𝐵))
71, 6impbid2 215 1 (Rel 𝐹 → (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195   = wceq 1475  ccnv 5037  Rel wrel 5043  1-1-ontowf1o 5803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811
This theorem is referenced by:  hasheqf1oi  13002  hasheqf1oiOLD  13003
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