Theorem fssd 5423

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 5422	. 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 3157   -->wf 5255

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170  df-f 5263

This theorem is referenced by:  mapss  6759  ac6sfi  6968  fseq1p1m1  10186  seqf1oglem2  10629  sswrd  10961  resqrexlemcvg  11201  resqrexlemsqa  11206  climcvg1nlem  11531  fsumcl2lem  11580  nninfctlemfo  12232  ennnfonelemh  12646  gsumress  13097  gsumwsubmcl  13198  gsumfzsubmcl  13544  cnrest2  14556  cnptoprest2  14560  cncfss  14903  limccnpcntop  14995  dvidre  15017  dvcoapbr  15027  dvef  15047  plyaddlem  15069  plymullem  15070  plycjlemc  15080  plycn  15082  dvply2g  15086  isomninnlem  15761  trilpolemisumle  15769  iswomninnlem  15780  ismkvnnlem  15783