Theorem fssd 5421

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 5420	. 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  6752  ac6sfi  6961  fseq1p1m1  10172  seqf1oglem2  10615  sswrd  10947  resqrexlemcvg  11187  resqrexlemsqa  11192  climcvg1nlem  11517  fsumcl2lem  11566  nninfctlemfo  12218  ennnfonelemh  12632  gsumress  13064  gsumwsubmcl  13154  gsumfzsubmcl  13494  cnrest2  14498  cnptoprest2  14502  cncfss  14845  limccnpcntop  14937  dvidre  14959  dvcoapbr  14969  dvef  14989  plyaddlem  15011  plymullem  15012  plycjlemc  15022  plycn  15024  dvply2g  15028  isomninnlem  15703  trilpolemisumle  15711  iswomninnlem  15722  ismkvnnlem  15725