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Theorem funres11 5080
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 4732 . 2  |-  ( F  |`  A )  C_  F
2 cnvss 4605 . 2  |-  ( ( F  |`  A )  C_  F  ->  `' ( F  |`  A )  C_  `' F )
3 funss 5028 . 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 2999   `'ccnv 4435    |` cres 4438   Fun wfun 5004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3005  df-ss 3012  df-br 3844  df-opab 3898  df-rel 4443  df-cnv 4444  df-co 4445  df-res 4448  df-fun 5012
This theorem is referenced by:  f1ssres  5219  resdif  5269  ssdomg  6485  sbthlemi8  6663
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