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Theorem funimass2 5570
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 5284 . 2  |-  ( A 
C_  ( `' F " B )  ->  ( F " A )  C_  ( F " ( `' F " B ) ) )
2 funimacnv 5568 . . . . 5  |-  ( Fun 
F  ->  ( F " ( `' F " B ) )  =  ( B  i^i  ran  F ) )
32sseq2d 3445 . . . 4  |-  ( Fun 
F  ->  ( ( F " A )  C_  ( F " ( `' F " B ) )  <->  ( F " A )  C_  ( B  i^i  ran  F )
) )
4 inss1 3632 . . . . 5  |-  ( B  i^i  ran  F )  C_  B
5 sstr2 3424 . . . . 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 228 . . 3  |-  ( Fun 
F  ->  ( ( F " A )  C_  ( F " ( `' F " B ) )  ->  ( F " A )  C_  B
) )
87imp 427 . 2  |-  ( ( Fun  F  /\  ( F " A )  C_  ( F " ( `' F " B ) ) )  ->  ( F " A )  C_  B )
91, 8sylan2 472 1  |-  ( ( Fun  F  /\  A  C_  ( `' F " B ) )  -> 
( F " A
)  C_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    i^i cin 3388    C_ wss 3389   `'ccnv 4912   ran crn 4914   "cima 4916   Fun wfun 5490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-fun 5498
This theorem is referenced by:  fvimacnvi  5903  lmhmlsp  17808  2ndcomap  20044  tgqtop  20298  kqreglem1  20327  fmfnfmlem4  20543  fmucnd  20880  cfilucfilOLD  21157  cfilucfil  21158
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