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Theorem funconstss 5285
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 5224 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  { B } 
<-> 
A. x  e.  A  ( F `  x )  e.  { B }
) )
2 funimass3 5283 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  { B } 
<->  A  C_  ( `' F " { B }
) ) )
3 ssel2 2940 . . . . . 6  |-  ( ( A  C_  dom  F  /\  x  e.  A )  ->  x  e.  dom  F
)
43anim2i 324 . . . . 5  |-  ( ( Fun  F  /\  ( A  C_  dom  F  /\  x  e.  A )
)  ->  ( Fun  F  /\  x  e.  dom  F ) )
54anassrs 380 . . . 4  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  x  e.  A
)  ->  ( Fun  F  /\  x  e.  dom  F ) )
6 funfvex 5192 . . . 4  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
7 elsng 3390 . . . 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 2322 . 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 208 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 97    <-> wb 98    = wceq 1243    e. wcel 1393   A.wral 2306   _Vcvv 2557    C_ wss 2917   {csn 3375   `'ccnv 4344   dom cdm 4345   "cima 4348   Fun wfun 4896   ` cfv 4902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910
This theorem is referenced by:  fconst3m  5380
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