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Theorem funconstss 5659
Description: Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.)
Assertion
Ref Expression
funconstss  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( A. x  e.  A  ( F `  x )  =  B  <-> 
A  C_  ( `' F " { B }
) ) )
Distinct variable groups:    x, F    x, A    x, B

Proof of Theorem funconstss
StepHypRef Expression
1 funimass4 5589 . . 3  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  { B } 
<-> 
A. x  e.  A  ( F `  x )  e.  { B }
) )
2 fvex 5555 . . . . 5  |-  ( F `
 x )  e. 
_V
32elsnc 3676 . . . 4  |-  ( ( F `  x )  e.  { B }  <->  ( F `  x )  =  B )
43ralbii 2580 . . 3  |-  ( A. x  e.  A  ( F `  x )  e.  { B }  <->  A. x  e.  A  ( F `  x )  =  B )
51, 4syl6rbb 253 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( A. x  e.  A  ( F `  x )  =  B  <-> 
( F " A
)  C_  { B } ) )
6 funimass3 5657 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  { B } 
<->  A  C_  ( `' F " { B }
) ) )
75, 6bitrd 244 1  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( A. x  e.  A  ( F `  x )  =  B  <-> 
A  C_  ( `' F " { B }
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   {csn 3653   `'ccnv 4704   dom cdm 4705   "cima 4708   Fun wfun 5265   ` cfv 5271
This theorem is referenced by:  fconst3  5751  ipasslem8  21431
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-sbc 3005  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-uni 3844  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-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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