ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fvss Unicode version

Theorem fvss 5401
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 5099 . 2  |-  ( F `
 A )  =  ( iota x A F x )
2 iotass 5073 . 2  |-  ( A. x ( A F x  ->  x  C_  B
)  ->  ( iota x A F x ) 
C_  B )
31, 2eqsstrid 3111 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 1312    C_ wss 3039   class class class wbr 3897   iotacio 5054   ` cfv 5091
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  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-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-uni 3705  df-iota 5056  df-fv 5099
This theorem is referenced by:  fvssunirng  5402  relfvssunirn  5403  sefvex  5408  fvmptss2  5462  tfrexlem  6197
  Copyright terms: Public domain W3C validator