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Theorem funres11 5163
Description: The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.)
Assertion
Ref Expression
funres11  |-  ( Fun  `' F  ->  Fun  `' ( F  |`  A ) )

Proof of Theorem funres11
StepHypRef Expression
1 resss 4811 . 2  |-  ( F  |`  A )  C_  F
2 cnvss 4680 . 2  |-  ( ( F  |`  A )  C_  F  ->  `' ( F  |`  A )  C_  `' F )
3 funss 5110 . 2  |-  ( `' ( F  |`  A ) 
C_  `' F  -> 
( Fun  `' F  ->  Fun  `' ( F  |`  A ) ) )
41, 2, 3mp2b 8 1  |-  ( Fun  `' F  ->  Fun  `' ( F  |`  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3039   `'ccnv 4506    |` cres 4509   Fun wfun 5085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045  df-ss 3052  df-br 3898  df-opab 3958  df-rel 4514  df-cnv 4515  df-co 4516  df-res 4519  df-fun 5093
This theorem is referenced by:  f1ssres  5305  resdif  5355  ssdomg  6638  sbthlemi8  6818
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