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

Theorem funimass5 5849
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 5848 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A 
C_  ( `' F " B ) ) )
2 funimass4 5779 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A. x  e.  A  ( F `  x )  e.  B ) )
31, 2bitr3d 248 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 178    /\ wa 360    e. wcel 1726   A.wral 2707    C_ wss 3322   `'ccnv 4879   dom cdm 4880   "cima 4883   Fun wfun 5450   ` cfv 5456
This theorem is referenced by:  clssubg  18140
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-fv 5464
  Copyright terms: Public domain W3C validator