Theorem fssd 5350

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 5349	. 2 |- ( ( F : A --> B /\ B C_ C ) -> F : A --> C )
4	1, 2, 3	syl2anc 409	1 |- ( ph -> F : A --> C )

Syntax hints:   -> wi 4   C_ wss 3116   -->wf 5184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129  df-f 5192

This theorem is referenced by:  mapss  6657  ac6sfi  6864  fseq1p1m1  10029  resqrexlemcvg  10961  resqrexlemsqa  10966  climcvg1nlem  11290  fsumcl2lem  11339  ennnfonelemh  12337  cnrest2  12876  cnptoprest2  12880  cncfss  13210  limccnpcntop  13284  dvcoapbr  13311  dvef  13328  isomninnlem  13909  trilpolemisumle  13917  iswomninnlem  13928  ismkvnnlem  13931