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Theorem fssd 5055
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 5054 . 2  |-  ( ( F : A --> B  /\  B  C_  C )  ->  F : A --> C )
41, 2, 3syl2anc 391 1  |-  ( ph  ->  F : A --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 2917   -->wf 4898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931  df-f 4906
This theorem is referenced by:  ac6sfi  6352  fseq1p1m1  8956  resqrexlemcvg  9617  resqrexlemsqa  9622  climcvg1nlem  9868
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