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Theorem funconstss 5504
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 5438 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  { B } 
<-> 
A. x  e.  A  ( F `  x )  e.  { B }
) )
2 funimass3 5502 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  { B } 
<->  A  C_  ( `' F " { B }
) ) )
3 ssel2 3060 . . . . . 6  |-  ( ( A  C_  dom  F  /\  x  e.  A )  ->  x  e.  dom  F
)
43anim2i 337 . . . . 5  |-  ( ( Fun  F  /\  ( A  C_  dom  F  /\  x  e.  A )
)  ->  ( Fun  F  /\  x  e.  dom  F ) )
54anassrs 395 . . . 4  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  x  e.  A
)  ->  ( Fun  F  /\  x  e.  dom  F ) )
6 funfvex 5404 . . . 4  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
7 elsng 3510 . . . 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 2408 . 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 1314    e. wcel 1463   A.wral 2391   _Vcvv 2658    C_ wss 3039   {csn 3495   `'ccnv 4506   dom cdm 4507   "cima 4510   Fun wfun 5085   ` cfv 5091
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099
This theorem is referenced by:  fconst3m  5605
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