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Theorem fvss 5220
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 4940 . 2  |-  ( F `
 A )  =  ( iota x A F x )
2 iotass 4914 . 2  |-  ( A. x ( A F x  ->  x  C_  B
)  ->  ( iota x A F x ) 
C_  B )
31, 2syl5eqss 3044 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 1283    C_ wss 2974   class class class wbr 3793   iotacio 4895   ` cfv 4932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-uni 3610  df-iota 4897  df-fv 4940
This theorem is referenced by:  fvssunirng  5221  relfvssunirn  5222  sefvex  5227  fvmptss2  5279  tfrexlem  5983
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