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Theorem f1ocnvb 6607
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 6606 . 2 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
2 f1ocnv 6606 . . 3 (𝐹:𝐵1-1-onto𝐴𝐹:𝐴1-1-onto𝐵)
3 dfrel2 6017 . . . 4 (Rel 𝐹𝐹 = 𝐹)
4 f1oeq1 6583 . . . 4 (𝐹 = 𝐹 → (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1-onto𝐵))
53, 4sylbi 220 . . 3 (Rel 𝐹 → (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1-onto𝐵))
62, 5syl5ib 247 . 2 (Rel 𝐹 → (𝐹:𝐵1-1-onto𝐴𝐹:𝐴1-1-onto𝐵))
71, 6impbid2 229 1 (Rel 𝐹 → (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  ccnv 5522  Rel wrel 5528  1-1-ontowf1o 6327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335
This theorem is referenced by:  hasheqf1oi  13712
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