Theorem funres11 5453

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 5087	. 2 |- ( F |` A ) C_ F
2		cnvss 4953	. 2 |- ( ( F |` A ) C_ F -> `' ( F |` A ) C_ `' F )
3		funss 5396	. 2 |- ( `' ( F |` A ) C_ `' F -> ( Fun `' F -> Fun `' ( F |` A ) ) )
4	1, 2, 3	mp2b 8	1 |- ( Fun `' F -> Fun `' ( F |` A ) )

This proof depends on syntax axioms:   -> wi 4   C_ wss 3220   `'ccnv 4773   |` cres 4776   Fun wfun 5371

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-br 4131  df-opab 4193  df-rel 4781  df-cnv 4782  df-co 4783  df-res 4786  df-fun 5379

This theorem is used by:  f1ssres  5607  resdif  5661  f1ssf1  5671  ssdomg  7065  sbthlemi8  7281