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Theorem funres11 5320
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 4979 . 2  |-  ( F  |`  A )  C_  F
2 cnvss 4854 . 2  |-  ( ( F  |`  A )  C_  F  ->  `' ( F  |`  A )  C_  `' F )
3 funss 5273 . 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 3152   `'ccnv 4688    |` cres 4691   Fun wfun 5249
This theorem is referenced by:  f1ssres  5444  resdif  5494  ssdomg  6907  sbthlem8  6978
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-res 4701  df-fun 5257
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