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Theorem f1cnv 5499
Description: The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)
Assertion
Ref Expression
f1cnv (𝐹:𝐴–1-1→𝐡 β†’ ◑𝐹:ran 𝐹–1-1-onto→𝐴)

Proof of Theorem f1cnv
StepHypRef Expression
1 f1f1orn 5486 . 2 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
2 f1ocnv 5488 . 2 (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 β†’ ◑𝐹:ran 𝐹–1-1-onto→𝐴)
31, 2syl 14 1 (𝐹:𝐴–1-1→𝐡 β†’ ◑𝐹:ran 𝐹–1-1-onto→𝐴)
Colors of variables: wff set class
Syntax hints:   β†’ wi 4  β—‘ccnv 4639  ran crn 4641  β€“1-1β†’wf1 5227  β€“1-1-ontoβ†’wf1o 5229
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-v 2753  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-br 4018  df-opab 4079  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237
This theorem is referenced by:  f1dmex  6134  f1dmvrnfibi  6960
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