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Theorem funimass1 5341
Description: A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by NM, 25-May-2004.)
Assertion
Ref Expression
funimass1  |-  ( ( Fun  F  /\  A  C_ 
ran  F )  -> 
( ( `' F " A )  C_  B  ->  A  C_  ( F " B ) ) )

Proof of Theorem funimass1
StepHypRef Expression
1 imass2 5065 . 2  |-  ( ( `' F " A ) 
C_  B  ->  ( F " ( `' F " A ) )  C_  ( F " B ) )
2 funimacnv 5340 . . . 4  |-  ( Fun 
F  ->  ( F " ( `' F " A ) )  =  ( A  i^i  ran  F ) )
3 dfss 3180 . . . . . 6  |-  ( A 
C_  ran  F  <->  A  =  ( A  i^i  ran  F
) )
43biimpi 186 . . . . 5  |-  ( A 
C_  ran  F  ->  A  =  ( A  i^i  ran 
F ) )
54eqcomd 2301 . . . 4  |-  ( A 
C_  ran  F  ->  ( A  i^i  ran  F
)  =  A )
62, 5sylan9eq 2348 . . 3  |-  ( ( Fun  F  /\  A  C_ 
ran  F )  -> 
( F " ( `' F " A ) )  =  A )
76sseq1d 3218 . 2  |-  ( ( Fun  F  /\  A  C_ 
ran  F )  -> 
( ( F "
( `' F " A ) )  C_  ( F " B )  <-> 
A  C_  ( F " B ) ) )
81, 7syl5ib 210 1  |-  ( ( Fun  F  /\  A  C_ 
ran  F )  -> 
( ( `' F " A )  C_  B  ->  A  C_  ( F " B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    i^i cin 3164    C_ wss 3165   `'ccnv 4704   ran crn 4706   "cima 4708   Fun wfun 5265
This theorem is referenced by:  kqnrmlem1  17450  hmeontr  17476  nrmhmph  17501  cnheiborlem  18468
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273
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