Theorem fssd 5527

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 5526	. 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 3214   -->wf 5353

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-in 3220  df-ss 3227  df-f 5361

This theorem is referenced by:  mapsnd  6936  mapss  6939  ac6sfi  7168  fseq1p1m1  10450  seqf1oglem2  10906  sswrd  11258  resqrexlemcvg  11729  resqrexlemsqa  11734  climcvg1nlem  12059  fsumcl2lem  12109  nninfctlemfo  12761  ennnfonelemh  13239  gsumress  13658  gsumwsubmcl  13751  gsumfzsubmcl  14091  cnrest2  15227  cnptoprest2  15231  cncfss  15574  limccnpcntop  15666  dvidre  15688  dvcoapbr  15698  dvef  15718  plyaddlem  15740  plymullem  15741  plycjlemc  15751  plycn  15753  dvply2g  15757  upgruhgr  16232  umgrupgr  16233  upgr1edc  16242  umgrislfupgrdom  16252  usgrislfuspgrdom  16311  isomninnlem  16940  trilpolemisumle  16948  iswomninnlem  16960  ismkvnnlem  16963