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Theorem funconstss 5851
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 5780 . . 3  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  { B } 
<-> 
A. x  e.  A  ( F `  x )  e.  { B }
) )
2 fvex 5745 . . . . 5  |-  ( F `
 x )  e. 
_V
32elsnc 3839 . . . 4  |-  ( ( F `  x )  e.  { B }  <->  ( F `  x )  =  B )
43ralbii 2731 . . 3  |-  ( A. x  e.  A  ( F `  x )  e.  { B }  <->  A. x  e.  A  ( F `  x )  =  B )
51, 4syl6rbb 255 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( A. x  e.  A  ( F `  x )  =  B  <-> 
( F " A
)  C_  { B } ) )
6 funimass3 5849 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  { B } 
<->  A  C_  ( `' F " { B }
) ) )
75, 6bitrd 246 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   {csn 3816   `'ccnv 4880   dom cdm 4881   "cima 4884   Fun wfun 5451   ` cfv 5457
This theorem is referenced by:  fconst3  5958  ipasslem8  22343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4333  ax-nul 4341  ax-pr 4406
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 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-fv 5465
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