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Theorem funimass2 5467
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 5180 . 2  |-  ( A 
C_  ( `' F " B )  ->  ( F " A )  C_  ( F " ( `' F " B ) ) )
2 funimacnv 5465 . . . . 5  |-  ( Fun 
F  ->  ( F " ( `' F " B ) )  =  ( B  i^i  ran  F ) )
32sseq2d 3319 . . . 4  |-  ( Fun 
F  ->  ( ( F " A )  C_  ( F " ( `' F " B ) )  <->  ( F " A )  C_  ( B  i^i  ran  F )
) )
4 inss1 3504 . . . . 5  |-  ( B  i^i  ran  F )  C_  B
5 sstr2 3298 . . . . 5  |-  ( ( F " A ) 
C_  ( B  i^i  ran 
F )  ->  (
( B  i^i  ran  F )  C_  B  ->  ( F " A ) 
C_  B ) )
64, 5mpi 17 . . . 4  |-  ( ( F " A ) 
C_  ( B  i^i  ran 
F )  ->  ( F " A )  C_  B )
73, 6syl6bi 220 . . 3  |-  ( Fun 
F  ->  ( ( F " A )  C_  ( F " ( `' F " B ) )  ->  ( F " A )  C_  B
) )
87imp 419 . 2  |-  ( ( Fun  F  /\  ( F " A )  C_  ( F " ( `' F " B ) ) )  ->  ( F " A )  C_  B )
91, 8sylan2 461 1  |-  ( ( Fun  F  /\  A  C_  ( `' F " B ) )  -> 
( F " A
)  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    i^i cin 3262    C_ wss 3263   `'ccnv 4817   ran crn 4819   "cima 4821   Fun wfun 5388
This theorem is referenced by:  fvimacnvi  5783  lmhmlsp  16052  2ndcomap  17442  tgqtop  17665  kqreglem1  17694  fmfnfmlem4  17910  fmucnd  18243  cfilucfil  18479
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-fun 5396
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