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Theorem funimass2 5326
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 5049 . 2  |-  ( A 
C_  ( `' F " B )  ->  ( F " A )  C_  ( F " ( `' F " B ) ) )
2 funimacnv 5324 . . . . 5  |-  ( Fun 
F  ->  ( F " ( `' F " B ) )  =  ( B  i^i  ran  F ) )
32sseq2d 3206 . . . 4  |-  ( Fun 
F  ->  ( ( F " A )  C_  ( F " ( `' F " B ) )  <->  ( F " A )  C_  ( B  i^i  ran  F )
) )
4 inss1 3389 . . . . 5  |-  ( B  i^i  ran  F )  C_  B
5 sstr2 3186 . . . . 5  |-  ( ( F " A ) 
C_  ( B  i^i  ran 
F )  ->  (
( B  i^i  ran  F )  C_  B  ->  ( F " A ) 
C_  B ) )
64, 5mpi 16 . . . 4  |-  ( ( F " A ) 
C_  ( B  i^i  ran 
F )  ->  ( F " A )  C_  B )
73, 6syl6bi 219 . . 3  |-  ( Fun 
F  ->  ( ( F " A )  C_  ( F " ( `' F " B ) )  ->  ( F " A )  C_  B
) )
87imp 418 . 2  |-  ( ( Fun  F  /\  ( F " A )  C_  ( F " ( `' F " B ) ) )  ->  ( F " A )  C_  B )
91, 8sylan2 460 1  |-  ( ( Fun  F  /\  A  C_  ( `' F " B ) )  -> 
( F " A
)  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    i^i cin 3151    C_ wss 3152   `'ccnv 4688   ran crn 4690   "cima 4692   Fun wfun 5249
This theorem is referenced by:  fvimacnvi  5639  lmhmlsp  15806  2ndcomap  17184  tgqtop  17403  kqreglem1  17432  fmfnfmlem4  17652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257
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