Theorem fssd 5495

Description: Expanding the codomain of a mapping, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
fssd.f	|- ( ph -> F : A --> B )
fssd.b	|- ( ph -> B C_ C )

Assertion

Ref	Expression
fssd	|- ( ph -> F : A --> C )

Proof of Theorem fssd

Step	Hyp	Ref	Expression
1		fssd.f	. 2 |- ( ph -> F : A --> B )
2		fssd.b	. 2 |- ( ph -> B C_ C )
3		fss 5494	. 2 |- ( ( F : A --> B /\ B C_ C ) -> F : A --> C )
4	1, 2, 3	syl2anc 411	1 |- ( ph -> F : A --> C )

Syntax hints:   -> wi 4   C_ wss 3200   -->wf 5322

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3206  df-ss 3213  df-f 5330

This theorem is referenced by:  mapss  6860  ac6sfi  7087  fseq1p1m1  10329  seqf1oglem2  10783  sswrd  11126  resqrexlemcvg  11597  resqrexlemsqa  11602  climcvg1nlem  11927  fsumcl2lem  11977  nninfctlemfo  12629  ennnfonelemh  13043  gsumress  13496  gsumwsubmcl  13597  gsumfzsubmcl  13943  cnrest2  14979  cnptoprest2  14983  cncfss  15326  limccnpcntop  15418  dvidre  15440  dvcoapbr  15450  dvef  15470  plyaddlem  15492  plymullem  15493  plycjlemc  15503  plycn  15505  dvply2g  15509  upgruhgr  15981  umgrupgr  15982  upgr1edc  15991  umgrislfupgrdom  16001  usgrislfuspgrdom  16060  isomninnlem  16685  trilpolemisumle  16693  iswomninnlem  16705  ismkvnnlem  16708