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Theorem funimass1 5643
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 5360 . 2  |-  ( ( `' F " A ) 
C_  B  ->  ( F " ( `' F " A ) )  C_  ( F " B ) )
2 funimacnv 5642 . . . 4  |-  ( Fun 
F  ->  ( F " ( `' F " A ) )  =  ( A  i^i  ran  F ) )
3 dfss 3476 . . . . . 6  |-  ( A 
C_  ran  F  <->  A  =  ( A  i^i  ran  F
) )
43biimpi 194 . . . . 5  |-  ( A 
C_  ran  F  ->  A  =  ( A  i^i  ran 
F ) )
54eqcomd 2462 . . . 4  |-  ( A 
C_  ran  F  ->  ( A  i^i  ran  F
)  =  A )
62, 5sylan9eq 2515 . . 3  |-  ( ( Fun  F  /\  A  C_ 
ran  F )  -> 
( F " ( `' F " A ) )  =  A )
76sseq1d 3516 . 2  |-  ( ( Fun  F  /\  A  C_ 
ran  F )  -> 
( ( F "
( `' F " A ) )  C_  ( F " B )  <-> 
A  C_  ( F " B ) ) )
81, 7syl5ib 219 1  |-  ( ( Fun  F  /\  A  C_ 
ran  F )  -> 
( ( `' F " A )  C_  B  ->  A  C_  ( F " B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    i^i cin 3460    C_ wss 3461   `'ccnv 4987   ran crn 4989   "cima 4991   Fun wfun 5564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-fun 5572
This theorem is referenced by:  kqnrmlem1  20410  hmeontr  20436  nrmhmph  20461  cnheiborlem  21620
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