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Theorem funimass2 5655
Description: A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by NM, 25-May-2004.)
Assertion
Ref Expression
funimass2  |-  ( ( Fun  F  /\  A  C_  ( `' F " B ) )  -> 
( F " A
)  C_  B )

Proof of Theorem funimass2
StepHypRef Expression
1 imass2 5203 . 2  |-  ( A 
C_  ( `' F " B )  ->  ( F " A )  C_  ( F " ( `' F " B ) ) )
2 funimacnv 5653 . . . . 5  |-  ( Fun 
F  ->  ( F " ( `' F " B ) )  =  ( B  i^i  ran  F ) )
32sseq2d 3459 . . . 4  |-  ( Fun 
F  ->  ( ( F " A )  C_  ( F " ( `' F " B ) )  <->  ( F " A )  C_  ( B  i^i  ran  F )
) )
4 inss1 3651 . . . . 5  |-  ( B  i^i  ran  F )  C_  B
5 sstr2 3438 . . . . 5  |-  ( ( F " A ) 
C_  ( B  i^i  ran 
F )  ->  (
( B  i^i  ran  F )  C_  B  ->  ( F " A ) 
C_  B ) )
64, 5mpi 20 . . . 4  |-  ( ( F " A ) 
C_  ( B  i^i  ran 
F )  ->  ( F " A )  C_  B )
73, 6syl6bi 232 . . 3  |-  ( Fun 
F  ->  ( ( F " A )  C_  ( F " ( `' F " B ) )  ->  ( F " A )  C_  B
) )
87imp 431 . 2  |-  ( ( Fun  F  /\  ( F " A )  C_  ( F " ( `' F " B ) ) )  ->  ( F " A )  C_  B )
91, 8sylan2 477 1  |-  ( ( Fun  F  /\  A  C_  ( `' F " B ) )  -> 
( F " A
)  C_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    i^i cin 3402    C_ wss 3403   `'ccnv 4832   ran crn 4834   "cima 4836   Fun wfun 5575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-br 4402  df-opab 4461  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-fun 5583
This theorem is referenced by:  fvimacnvi  5994  lmhmlsp  18265  2ndcomap  20466  tgqtop  20720  kqreglem1  20749  fmfnfmlem4  20965  fmucnd  21300  cfilucfil  21567
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