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Theorem f1cnv 6323
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 6311 . 2 (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)
2 f1ocnv 6312 . 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 5266  ran crn 5268  1-1wf1 6047  1-1-ontowf1o 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057
This theorem is referenced by:  f1dmex  7303  fin1a2lem7  9441  diophrw  37843
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