Theorem fssd 5438

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 5437	. 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 3166   -->wf 5267

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-in 3172  df-ss 3179  df-f 5275

This theorem is referenced by:  mapss  6778  ac6sfi  6995  fseq1p1m1  10216  seqf1oglem2  10665  sswrd  11003  resqrexlemcvg  11330  resqrexlemsqa  11335  climcvg1nlem  11660  fsumcl2lem  11709  nninfctlemfo  12361  ennnfonelemh  12775  gsumress  13227  gsumwsubmcl  13328  gsumfzsubmcl  13674  cnrest2  14708  cnptoprest2  14712  cncfss  15055  limccnpcntop  15147  dvidre  15169  dvcoapbr  15179  dvef  15199  plyaddlem  15221  plymullem  15222  plycjlemc  15232  plycn  15234  dvply2g  15238  isomninnlem  15969  trilpolemisumle  15977  iswomninnlem  15988  ismkvnnlem  15991