Theorem fssd 5417

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 5416	. 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 3154   -->wf 5251

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3160  df-ss 3167  df-f 5259

This theorem is referenced by:  mapss  6747  ac6sfi  6956  fseq1p1m1  10163  seqf1oglem2  10594  sswrd  10926  resqrexlemcvg  11166  resqrexlemsqa  11171  climcvg1nlem  11495  fsumcl2lem  11544  nninfctlemfo  12180  ennnfonelemh  12564  gsumress  12981  gsumwsubmcl  13071  gsumfzsubmcl  13411  cnrest2  14415  cnptoprest2  14419  cncfss  14762  limccnpcntop  14854  dvidre  14876  dvcoapbr  14886  dvef  14906  plyaddlem  14928  plymullem  14929  plycjlemc  14938  plycn  14940  isomninnlem  15590  trilpolemisumle  15598  iswomninnlem  15609  ismkvnnlem  15612