Theorem fssd 5360

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 5359	. 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 3121   -->wf 5194

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134  df-f 5202

This theorem is referenced by:  mapss  6669  ac6sfi  6876  fseq1p1m1  10050  resqrexlemcvg  10983  resqrexlemsqa  10988  climcvg1nlem  11312  fsumcl2lem  11361  ennnfonelemh  12359  cnrest2  13030  cnptoprest2  13034  cncfss  13364  limccnpcntop  13438  dvcoapbr  13465  dvef  13482  isomninnlem  14062  trilpolemisumle  14070  iswomninnlem  14081  ismkvnnlem  14084