Theorem fss 5521

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 3245	. . . . 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 5356	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
5		df-f 5356	. . 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 3211   ran crn 4750   Fn wfn 5347   -->wf 5348

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 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-in 3217  df-ss 3224  df-f 5356

This theorem is referenced by:  fssd  5522  fconst6g  5566  f1ss  5579  ffoss  5647  fsn2  5851  ofco  6285  tposf2  6499  issmo2  6520  smoiso  6533  mapsn  6925  ssdomg  7018  omp1eomlem  7385  1fv  10473  fxnn0nninf  10801  abscn2  12000  recn2  12002  imcn2  12003  climabs  12005  climre  12007  climim  12008  fsumre  12158  fsumim  12159  resmhm2  13701  prdsgrpd  13822  prdsinvgd  13823  ismet2  15219  dvfre  15575  dvrecap  15578  elplyr  15605  lgsfcl  15881  konigsbergssiedgwen  16481