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Theorem funres11 5336
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 4995 . 2  |-  ( F  |`  A )  C_  F
2 cnvss 4870 . 2  |-  ( ( F  |`  A )  C_  F  ->  `' ( F  |`  A )  C_  `' F )
3 funss 5289 . 2  |-  ( `' ( F  |`  A ) 
C_  `' F  -> 
( Fun  `' F  ->  Fun  `' ( F  |`  A ) ) )
41, 2, 3mp2b 9 1  |-  ( Fun  `' F  ->  Fun  `' ( F  |`  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3165   `'ccnv 4704    |` cres 4707   Fun wfun 5265
This theorem is referenced by:  f1ssres  5460  resdif  5510  ssdomg  6923  sbthlem8  6994  spthispth  28359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-ss 3179  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-res 4717  df-fun 5273
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