Theorem fssd 5416

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 5415	. 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 3153   -->wf 5250

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 3159  df-ss 3166  df-f 5258

This theorem is referenced by:  mapss  6745  ac6sfi  6954  fseq1p1m1  10160  seqf1oglem2  10591  sswrd  10923  resqrexlemcvg  11163  resqrexlemsqa  11168  climcvg1nlem  11492  fsumcl2lem  11541  nninfctlemfo  12177  ennnfonelemh  12561  gsumress  12978  gsumwsubmcl  13068  gsumfzsubmcl  13408  cnrest2  14404  cnptoprest2  14408  cncfss  14738  limccnpcntop  14829  dvcoapbr  14856  dvef  14873  plyaddlem  14895  plymullem  14896  isomninnlem  15520  trilpolemisumle  15528  iswomninnlem  15539  ismkvnnlem  15542