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  10546  fxnn0nninf  10876  hashf1lem1  11285  abscn2  12081  recn2  12083  imcn2  12084  climabs  12086  climre  12088  climim  12089  fsumre  12239  fsumim  12240  resmhm2  13795  prdsgrpd  14197  prdsinvgd  14198  ismet2  15455  dvfre  15811  dvrecap  15814  elplyr  15841  lgsfcl  16127  konigsbergssiedgwen  16727