Theorem funres11 5966

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 5422	. 2 |- ( F |` A ) C_ F
2		cnvss 5294	. 2 |- ( ( F |` A ) C_ F -> `' ( F |` A ) C_ `' F )
3		funss 5907	. 2 |- ( `' ( F |` A ) C_ `' F -> ( Fun `' F -> Fun `' ( F |` A ) ) )
4	1, 2, 3	mp2b 10	1 |- ( Fun `' F -> Fun `' ( F |` A ) )

Syntax hints:   -> wi 4   C_ wss 3574   `'ccnv 5113   |` cres 5116   Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-res 5126  df-fun 5890

This theorem is referenced by:  f1ssres  6108  resdif  6157  f1ssf1  6168  ssdomg  8001  sbthlem8  8077  spthispth  26622