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Theorem funres11 5522
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 5171 . 2  |-  ( F  |`  A )  C_  F
2 cnvss 5046 . 2  |-  ( ( F  |`  A )  C_  F  ->  `' ( F  |`  A )  C_  `' F )
3 funss 5473 . 2  |-  ( `' ( F  |`  A ) 
C_  `' F  -> 
( Fun  `' F  ->  Fun  `' ( F  |`  A ) ) )
41, 2, 3mp2b 10 1  |-  ( Fun  `' F  ->  Fun  `' ( F  |`  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3321   `'ccnv 4878    |` cres 4881   Fun wfun 5449
This theorem is referenced by:  f1ssres  5647  resdif  5697  ssdomg  7154  sbthlem8  7225  spthispth  21574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-v 2959  df-in 3328  df-ss 3335  df-br 4214  df-opab 4268  df-rel 4886  df-cnv 4887  df-co 4888  df-res 4891  df-fun 5457
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