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Theorem funimass5 5832
Description: A subclass of a preimage in terms of function values. (Contributed by NM, 15-May-2007.)
Assertion
Ref Expression
funimass5  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( A  C_  ( `' F " B )  <->  A. x  e.  A  ( F `  x )  e.  B ) )
Distinct variable groups:    x, F    x, A    x, B

Proof of Theorem funimass5
StepHypRef Expression
1 funimass3 5831 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A 
C_  ( `' F " B ) ) )
2 funimass4 5754 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A. x  e.  A  ( F `  x )  e.  B ) )
31, 2bitr3d 255 1  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( A  C_  ( `' F " B )  <->  A. x  e.  A  ( F `  x )  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1756   A.wral 2727    C_ wss 3340   `'ccnv 4851   dom cdm 4852   "cima 4855   Fun wfun 5424   ` cfv 5430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-fv 5438
This theorem is referenced by:  clssubg  19691
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