Theorem fss 5501

Description: Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
fss	|- ( ( F : A --> B /\ B C_ C ) -> F : A --> C )

Proof of Theorem fss

Step	Hyp	Ref	Expression
1		sstr2 3235	. . . . 5 |- ( ran F C_ B -> ( B C_ C -> ran F C_ C ) )
2	1	com12 30	. . . 4 |- ( B C_ C -> ( ran F C_ B -> ran F C_ C ) )
3	2	anim2d 337	. . 3 |- ( B C_ C -> ( ( F Fn A /\ ran F C_ B ) -> ( F Fn A /\ ran F C_ C ) ) )
4		df-f 5337	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
5		df-f 5337	. . 3 |- ( F : A --> C <-> ( F Fn A /\ ran F C_ C ) )
6	3, 4, 5	3imtr4g 205	. 2 |- ( B C_ C -> ( F : A --> B -> F : A --> C ) )
7	6	impcom 125	1 |- ( ( F : A --> B /\ B C_ C ) -> F : A --> C )

Syntax hints:   -> wi 4   /\ wa 104   C_ wss 3201   ran crn 4732   Fn wfn 5328   -->wf 5329

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-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3207  df-ss 3214  df-f 5337

This theorem is referenced by:  fssd  5502  fconst6g  5544  f1ss  5557  ffoss  5625  fsn2  5829  ofco  6263  tposf2  6477  issmo2  6498  smoiso  6511  mapsn  6902  ssdomg  6995  omp1eomlem  7336  1fv  10419  fxnn0nninf  10747  abscn2  11938  recn2  11940  imcn2  11941  climabs  11943  climre  11945  climim  11946  fsumre  12096  fsumim  12097  resmhm2  13634  prdsgrpd  13755  prdsinvgd  13756  ismet2  15148  dvfre  15504  dvrecap  15507  elplyr  15534  lgsfcl  15810  konigsbergssiedgwen  16410