Theorem fss 5546

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 3255	. . . . 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 5381	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
5		df-f 5381	. . 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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   C_ wss 3220   ran crn 4775   Fn wfn 5372   -->wf 5373

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233  df-f 5381

This theorem is used by:  fssd  5547  fconst6g  5591  f1ss  5604  ffoss  5672  fsn2  5882  ofco  6321  tposf2  6539  issmo2  6560  smoiso  6573  mapsn  6972  ssdomg  7065  omp1eomlem  7434  1fv  10556  fxnn0nninf  10889  hashf1lem1  11299  abscn2  12097  recn2  12099  imcn2  12100  climabs  12102  climre  12104  climim  12105  fsumre  12255  fsumim  12256  resmhm2  13844  prdsgrpd  14246  prdsinvgd  14247  ismet2  15504  dvfre  15860  dvrecap  15863  elplyr  15890  lgsfcl  16225  konigsbergssiedgwen  16825