Theorem fssd 5243

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 5242	. 2 |- ( ( F : A --> B /\ B C_ C ) -> F : A --> C )
4	1, 2, 3	syl2anc 406	1 |- ( ph -> F : A --> C )

Syntax hints:   -> wi 4   C_ wss 3037   -->wf 5077

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-in 3043  df-ss 3050  df-f 5085

This theorem is referenced by:  mapss  6539  ac6sfi  6745  fseq1p1m1  9767  resqrexlemcvg  10683  resqrexlemsqa  10688  climcvg1nlem  11010  fsumcl2lem  11059  ennnfonelemh  11762  cnrest2  12247  cnptoprest2  12251  cncfss  12556  limccnpcntop  12600  isomninnlem  12915  trilpolemisumle  12921