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Theorem funimass1 5091
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 4808 . 2  |-  ( ( `' F " A ) 
C_  B  ->  ( F " ( `' F " A ) )  C_  ( F " B ) )
2 funimacnv 5090 . . . 4  |-  ( Fun 
F  ->  ( F " ( `' F " A ) )  =  ( A  i^i  ran  F ) )
3 dfss 3013 . . . . . 6  |-  ( A 
C_  ran  F  <->  A  =  ( A  i^i  ran  F
) )
43biimpi 118 . . . . 5  |-  ( A 
C_  ran  F  ->  A  =  ( A  i^i  ran 
F ) )
54eqcomd 2093 . . . 4  |-  ( A 
C_  ran  F  ->  ( A  i^i  ran  F
)  =  A )
62, 5sylan9eq 2140 . . 3  |-  ( ( Fun  F  /\  A  C_ 
ran  F )  -> 
( F " ( `' F " A ) )  =  A )
76sseq1d 3053 . 2  |-  ( ( Fun  F  /\  A  C_ 
ran  F )  -> 
( ( F "
( `' F " A ) )  C_  ( F " B )  <-> 
A  C_  ( F " B ) ) )
81, 7syl5ib 152 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 102    = wceq 1289    i^i cin 2998    C_ wss 2999   `'ccnv 4437   ran crn 4439   "cima 4441   Fun wfun 5009
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-fun 5017
This theorem is referenced by: (None)
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