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Theorem f1cnv 6379
Description: The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)
Assertion
Ref Expression
f1cnv (𝐹:𝐴1-1𝐵𝐹:ran 𝐹1-1-onto𝐴)

Proof of Theorem f1cnv
StepHypRef Expression
1 f1f1orn 6367 . 2 (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)
2 f1ocnv 6368 . 2 (𝐹:𝐴1-1-onto→ran 𝐹𝐹:ran 𝐹1-1-onto𝐴)
31, 2syl 17 1 (𝐹:𝐴1-1𝐵𝐹:ran 𝐹1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  ccnv 5311  ran crn 5313  1-1wf1 6098  1-1-ontowf1o 6100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108
This theorem is referenced by:  f1dmex  7371  fin1a2lem7  9516  diophrw  38108
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