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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
StepHypRef 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 )
41, 2, 3syl2anc 406 1  |-  ( ph  ->  F : A --> C )
Colors of variables: wff set class
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
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