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Theorem f1ocnvb 5140
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 5139 . 2 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
2 f1ocnv 5139 . . 3 (𝐹:𝐵1-1-onto𝐴𝐹:𝐴1-1-onto𝐵)
3 dfrel2 4771 . . . 4 (Rel 𝐹𝐹 = 𝐹)
4 f1oeq1 5117 . . . 4 (𝐹 = 𝐹 → (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1-onto𝐵))
53, 4sylbi 114 . . 3 (Rel 𝐹 → (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1-onto𝐵))
62, 5syl5ib 143 . 2 (Rel 𝐹 → (𝐹:𝐵1-1-onto𝐴𝐹:𝐴1-1-onto𝐵))
71, 6impbid2 131 1 (Rel 𝐹 → (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1243  ccnv 4344  Rel wrel 4350  1-1-ontowf1o 4901
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909
This theorem is referenced by: (None)
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