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Theorem f1ocnvb 5497
Description: A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and codomain/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 5496 . 2 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
2 f1ocnv 5496 . . 3 (𝐹:𝐵1-1-onto𝐴𝐹:𝐴1-1-onto𝐵)
3 dfrel2 5100 . . . 4 (Rel 𝐹𝐹 = 𝐹)
4 f1oeq1 5471 . . . 4 (𝐹 = 𝐹 → (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1-onto𝐵))
53, 4sylbi 121 . . 3 (Rel 𝐹 → (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1-onto𝐵))
62, 5imbitrid 154 . 2 (Rel 𝐹 → (𝐹:𝐵1-1-onto𝐴𝐹:𝐴1-1-onto𝐵))
71, 6impbid2 143 1 (Rel 𝐹 → (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1364  ccnv 4646  Rel wrel 4652  1-1-ontowf1o 5237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-br 4022  df-opab 4083  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245
This theorem is referenced by: (None)
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