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Theorem funconstss 5506
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 5440 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  { B } 
<-> 
A. x  e.  A  ( F `  x )  e.  { B }
) )
2 funimass3 5504 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  { B } 
<->  A  C_  ( `' F " { B }
) ) )
3 ssel2 3062 . . . . . 6  |-  ( ( A  C_  dom  F  /\  x  e.  A )  ->  x  e.  dom  F
)
43anim2i 339 . . . . 5  |-  ( ( Fun  F  /\  ( A  C_  dom  F  /\  x  e.  A )
)  ->  ( Fun  F  /\  x  e.  dom  F ) )
54anassrs 397 . . . 4  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  x  e.  A
)  ->  ( Fun  F  /\  x  e.  dom  F ) )
6 funfvex 5406 . . . 4  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
7 elsng 3512 . . . 4  |-  ( ( F `  x )  e.  _V  ->  (
( F `  x
)  e.  { B } 
<->  ( F `  x
)  =  B ) )
85, 6, 73syl 17 . . 3  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  x  e.  A
)  ->  ( ( F `  x )  e.  { B }  <->  ( F `  x )  =  B ) )
98ralbidva 2410 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( A. x  e.  A  ( F `  x )  e.  { B }  <->  A. x  e.  A  ( F `  x )  =  B ) )
101, 2, 93bitr3rd 218 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    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   A.wral 2393   _Vcvv 2660    C_ wss 3041   {csn 3497   `'ccnv 4508   dom cdm 4509   "cima 4512   Fun wfun 5087   ` cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101
This theorem is referenced by:  fconst3m  5607
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