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Theorem funimass2 5519
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 5232 . 2  |-  ( A 
C_  ( `' F " B )  ->  ( F " A )  C_  ( F " ( `' F " B ) ) )
2 funimacnv 5517 . . . . 5  |-  ( Fun 
F  ->  ( F " ( `' F " B ) )  =  ( B  i^i  ran  F ) )
32sseq2d 3368 . . . 4  |-  ( Fun 
F  ->  ( ( F " A )  C_  ( F " ( `' F " B ) )  <->  ( F " A )  C_  ( B  i^i  ran  F )
) )
4 inss1 3553 . . . . 5  |-  ( B  i^i  ran  F )  C_  B
5 sstr2 3347 . . . . 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 3311    C_ wss 3312   `'ccnv 4869   ran crn 4871   "cima 4873   Fun wfun 5440
This theorem is referenced by:  fvimacnvi  5836  lmhmlsp  16117  2ndcomap  17513  tgqtop  17736  kqreglem1  17765  fmfnfmlem4  17981  fmucnd  18314  cfilucfilOLD  18591  cfilucfil  18592
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-fun 5448
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