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Theorem funimass1 5200
Description: A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by NM, 25-May-2004.)
Assertion
Ref Expression
funimass1  |-  ( ( Fun  F  /\  A  C_ 
ran  F )  -> 
( ( `' F " A )  C_  B  ->  A  C_  ( F " B ) ) )

Proof of Theorem funimass1
StepHypRef Expression
1 imass2 4915 . 2  |-  ( ( `' F " A ) 
C_  B  ->  ( F " ( `' F " A ) )  C_  ( F " B ) )
2 funimacnv 5199 . . . 4  |-  ( Fun 
F  ->  ( F " ( `' F " A ) )  =  ( A  i^i  ran  F ) )
3 dfss 3085 . . . . . 6  |-  ( A 
C_  ran  F  <->  A  =  ( A  i^i  ran  F
) )
43biimpi 119 . . . . 5  |-  ( A 
C_  ran  F  ->  A  =  ( A  i^i  ran 
F ) )
54eqcomd 2145 . . . 4  |-  ( A 
C_  ran  F  ->  ( A  i^i  ran  F
)  =  A )
62, 5sylan9eq 2192 . . 3  |-  ( ( Fun  F  /\  A  C_ 
ran  F )  -> 
( F " ( `' F " A ) )  =  A )
76sseq1d 3126 . 2  |-  ( ( Fun  F  /\  A  C_ 
ran  F )  -> 
( ( F "
( `' F " A ) )  C_  ( F " B )  <-> 
A  C_  ( F " B ) ) )
81, 7syl5ib 153 1  |-  ( ( Fun  F  /\  A  C_ 
ran  F )  -> 
( ( `' F " A )  C_  B  ->  A  C_  ( F " B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    i^i cin 3070    C_ wss 3071   `'ccnv 4538   ran crn 4540   "cima 4542   Fun wfun 5117
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-fun 5125
This theorem is referenced by:  hmeontr  12491
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