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Theorem f1ocnvdm 5795
Description: The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.)
Assertion
Ref Expression
f1ocnvdm  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  ( `' F `  C )  e.  A
)

Proof of Theorem f1ocnvdm
StepHypRef Expression
1 f1ocnv 5486 . . 3  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
2 f1of 5473 . . 3  |-  ( `' F : B -1-1-onto-> A  ->  `' F : B --> A )
31, 2syl 14 . 2  |-  ( F : A -1-1-onto-> B  ->  `' F : B --> A )
43ffvelcdmda 5664 1  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  ( `' F `  C )  e.  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2158   `'ccnv 4637   -->wf 5224   -1-1-onto->wf1o 5227   ` cfv 5228
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 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236
This theorem is referenced by:  f1oiso2  5841  f1ocnvfv3  5877  frecuzrdglem  10424  frecuzrdgtcl  10425  frecuzrdgsuc  10427  frecuzrdgdomlem  10430  frecuzrdgfunlem  10432  frecuzrdgsuctlem  10436  frecfzennn  10439  fzfig  10443
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