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Theorem fvss 5483
Description: The value of a function is a subset of  B if every element that could be a candidate for the value is a subset of  B. (Contributed by Mario Carneiro, 24-May-2019.)
Assertion
Ref Expression
fvss  |-  ( A. x ( A F x  ->  x  C_  B
)  ->  ( F `  A )  C_  B
)
Distinct variable groups:    x, A    x, B    x, F

Proof of Theorem fvss
StepHypRef Expression
1 df-fv 5179 . 2  |-  ( F `
 A )  =  ( iota x A F x )
2 iotass 5153 . 2  |-  ( A. x ( A F x  ->  x  C_  B
)  ->  ( iota x A F x ) 
C_  B )
31, 2eqsstrid 3174 1  |-  ( A. x ( A F x  ->  x  C_  B
)  ->  ( F `  A )  C_  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1333    C_ wss 3102   class class class wbr 3966   iotacio 5134   ` cfv 5171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3774  df-iota 5136  df-fv 5179
This theorem is referenced by:  fvssunirng  5484  relfvssunirn  5485  sefvex  5490  fvmptss2  5544  tfrexlem  6282
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