Theorem funres11 5326

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

Step	Hyp	Ref	Expression
1		resss 4966	. 2 |- ( F |` A ) C_ F
2		cnvss 4835	. 2 |- ( ( F |` A ) C_ F -> `' ( F |` A ) C_ `' F )
3		funss 5273	. 2 |- ( `' ( F |` A ) C_ `' F -> ( Fun `' F -> Fun `' ( F |` A ) ) )
4	1, 2, 3	mp2b 8	1 |- ( Fun `' F -> Fun `' ( F |` A ) )

Syntax hints:   -> wi 4   C_ wss 3153   `'ccnv 4658   |` cres 4661   Fun wfun 5248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159  df-ss 3166  df-br 4030  df-opab 4091  df-rel 4666  df-cnv 4667  df-co 4668  df-res 4671  df-fun 5256

This theorem is referenced by:  f1ssres  5468  resdif  5522  ssdomg  6832  sbthlemi8  7023