Theorem fss 5284

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 3104	. . . . 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 335	. . 3 |- ( B C_ C -> ( ( F Fn A /\ ran F C_ B ) -> ( F Fn A /\ ran F C_ C ) ) )
4		df-f 5127	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
5		df-f 5127	. . 3 |- ( F : A --> C <-> ( F Fn A /\ ran F C_ C ) )
6	3, 4, 5	3imtr4g 204	. 2 |- ( B C_ C -> ( F : A --> B -> F : A --> C ) )
7	6	impcom 124	1 |- ( ( F : A --> B /\ B C_ C ) -> F : A --> C )

Syntax hints:   -> wi 4   /\ wa 103   C_ wss 3071   ran crn 4540   Fn wfn 5118   -->wf 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084  df-f 5127

This theorem is referenced by:  fssd  5285  fconst6g  5321  f1ss  5334  ffoss  5399  fsn2  5594  ofco  6000  tposf2  6165  issmo2  6186  smoiso  6199  mapsn  6584  ssdomg  6672  omp1eomlem  6979  1fv  9916  fxnn0nninf  10211  abscn2  11084  recn2  11086  imcn2  11087  climabs  11089  climre  11091  climim  11092  fsumre  11241  fsumim  11242  ismet2  12523  dvfre  12843  dvrecap  12846