Theorem funres11 5330

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 4970	. 2 |- ( F |` A ) C_ F
2		cnvss 4839	. 2 |- ( ( F |` A ) C_ F -> `' ( F |` A ) C_ `' F )
3		funss 5277	. 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 3157   `'ccnv 4662   |` cres 4665   Fun wfun 5252

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-ss 3170  df-br 4034  df-opab 4095  df-rel 4670  df-cnv 4671  df-co 4672  df-res 4675  df-fun 5260

This theorem is referenced by:  f1ssres  5472  resdif  5526  ssdomg  6837  sbthlemi8  7030