Theorem funres11 5365

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 5002	. 2 |- ( F |` A ) C_ F
2		cnvss 4869	. 2 |- ( ( F |` A ) C_ F -> `' ( F |` A ) C_ `' F )
3		funss 5309	. 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 3174   `'ccnv 4692   |` cres 4695   Fun wfun 5284

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-ss 3187  df-br 4060  df-opab 4122  df-rel 4700  df-cnv 4701  df-co 4702  df-res 4705  df-fun 5292

This theorem is referenced by:  f1ssres  5512  resdif  5566  ssdomg  6893  sbthlemi8  7092