Theorem fss 5457

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 3208	. . . . 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 5294	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
5		df-f 5294	. . 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 3174   ran crn 4694   Fn wfn 5285   -->wf 5286

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-in 3180  df-ss 3187  df-f 5294

This theorem is referenced by:  fssd  5458  fconst6g  5496  f1ss  5509  ffoss  5576  fsn2  5777  ofco  6200  tposf2  6377  issmo2  6398  smoiso  6411  mapsn  6800  ssdomg  6893  omp1eomlem  7222  1fv  10296  fxnn0nninf  10621  abscn2  11741  recn2  11743  imcn2  11744  climabs  11746  climre  11748  climim  11749  fsumre  11898  fsumim  11899  resmhm2  13435  prdsgrpd  13556  prdsinvgd  13557  ismet2  14941  dvfre  15297  dvrecap  15300  elplyr  15327  lgsfcl  15600