MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funimass5 Unicode version

Theorem funimass5 5810
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 5809 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A 
C_  ( `' F " B ) ) )
2 funimass4 5740 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A. x  e.  A  ( F `  x )  e.  B ) )
31, 2bitr3d 247 1  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( A  C_  ( `' F " B )  <->  A. x  e.  A  ( F `  x )  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1721   A.wral 2670    C_ wss 3284   `'ccnv 4840   dom cdm 4841   "cima 4844   Fun wfun 5411   ` cfv 5417
This theorem is referenced by:  clssubg  18095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-fv 5425
  Copyright terms: Public domain W3C validator