Theorem fss 5485

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 3231	. . . . 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 5322	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
5		df-f 5322	. . 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 3197   ran crn 4720   Fn wfn 5313   -->wf 5314

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210  df-f 5322

This theorem is referenced by:  fssd  5486  fconst6g  5526  f1ss  5539  ffoss  5606  fsn2  5811  ofco  6243  tposf2  6420  issmo2  6441  smoiso  6454  mapsn  6845  ssdomg  6938  omp1eomlem  7272  1fv  10347  fxnn0nninf  10673  abscn2  11841  recn2  11843  imcn2  11844  climabs  11846  climre  11848  climim  11849  fsumre  11998  fsumim  11999  resmhm2  13536  prdsgrpd  13657  prdsinvgd  13658  ismet2  15043  dvfre  15399  dvrecap  15402  elplyr  15429  lgsfcl  15702