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Theorem funimass1 5667
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 5378 . 2  |-  ( ( `' F " A ) 
C_  B  ->  ( F " ( `' F " A ) )  C_  ( F " B ) )
2 funimacnv 5666 . . . 4  |-  ( Fun 
F  ->  ( F " ( `' F " A ) )  =  ( A  i^i  ran  F ) )
3 dfss 3496 . . . . . 6  |-  ( A 
C_  ran  F  <->  A  =  ( A  i^i  ran  F
) )
43biimpi 194 . . . . 5  |-  ( A 
C_  ran  F  ->  A  =  ( A  i^i  ran 
F ) )
54eqcomd 2475 . . . 4  |-  ( A 
C_  ran  F  ->  ( A  i^i  ran  F
)  =  A )
62, 5sylan9eq 2528 . . 3  |-  ( ( Fun  F  /\  A  C_ 
ran  F )  -> 
( F " ( `' F " A ) )  =  A )
76sseq1d 3536 . 2  |-  ( ( Fun  F  /\  A  C_ 
ran  F )  -> 
( ( F "
( `' F " A ) )  C_  ( F " B )  <-> 
A  C_  ( F " B ) ) )
81, 7syl5ib 219 1  |-  ( ( Fun  F  /\  A  C_ 
ran  F )  -> 
( ( `' F " A )  C_  B  ->  A  C_  ( F " B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    i^i cin 3480    C_ wss 3481   `'ccnv 5004   ran crn 5006   "cima 5008   Fun wfun 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-fun 5596
This theorem is referenced by:  kqnrmlem1  20112  hmeontr  20138  nrmhmph  20163  cnheiborlem  21322
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