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Theorem fsuppeq 27362
Description: Two ways of writing the support of a function with known codomain. MOVABLE SHORTEN nn0supp (Contributed by Stefan O'Rear, 9-Jul-2015.)
Assertion
Ref Expression
fsuppeq  |-  ( F : I --> S  -> 
( `' F "
( _V  \  { X } ) )  =  ( `' F "
( S  \  { X } ) ) )

Proof of Theorem fsuppeq
StepHypRef Expression
1 invdif 3423 . . 3  |-  ( S  i^i  ( _V  \  { X } ) )  =  ( S  \  { X } )
21imaeq2i 5026 . 2  |-  ( `' F " ( S  i^i  ( _V  \  { X } ) ) )  =  ( `' F " ( S 
\  { X }
) )
3 ffun 5407 . . . 4  |-  ( F : I --> S  ->  Fun  F )
4 inpreima 5668 . . . 4  |-  ( Fun 
F  ->  ( `' F " ( S  i^i  ( _V  \  { X } ) ) )  =  ( ( `' F " S )  i^i  ( `' F " ( _V  \  { X } ) ) ) )
53, 4syl 15 . . 3  |-  ( F : I --> S  -> 
( `' F "
( S  i^i  ( _V  \  { X }
) ) )  =  ( ( `' F " S )  i^i  ( `' F " ( _V 
\  { X }
) ) ) )
6 cnvimass 5049 . . . . 5  |-  ( `' F " ( _V 
\  { X }
) )  C_  dom  F
7 fdm 5409 . . . . . 6  |-  ( F : I --> S  ->  dom  F  =  I )
8 fimacnv 5673 . . . . . 6  |-  ( F : I --> S  -> 
( `' F " S )  =  I )
97, 8eqtr4d 2331 . . . . 5  |-  ( F : I --> S  ->  dom  F  =  ( `' F " S ) )
106, 9syl5sseq 3239 . . . 4  |-  ( F : I --> S  -> 
( `' F "
( _V  \  { X } ) )  C_  ( `' F " S ) )
11 sseqin2 3401 . . . 4  |-  ( ( `' F " ( _V 
\  { X }
) )  C_  ( `' F " S )  <-> 
( ( `' F " S )  i^i  ( `' F " ( _V 
\  { X }
) ) )  =  ( `' F "
( _V  \  { X } ) ) )
1210, 11sylib 188 . . 3  |-  ( F : I --> S  -> 
( ( `' F " S )  i^i  ( `' F " ( _V 
\  { X }
) ) )  =  ( `' F "
( _V  \  { X } ) ) )
135, 12eqtrd 2328 . 2  |-  ( F : I --> S  -> 
( `' F "
( S  i^i  ( _V  \  { X }
) ) )  =  ( `' F "
( _V  \  { X } ) ) )
142, 13syl5reqr 2343 1  |-  ( F : I --> S  -> 
( `' F "
( _V  \  { X } ) )  =  ( `' F "
( S  \  { X } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165   {csn 3653   `'ccnv 4704   dom cdm 4705   "cima 4708   Fun wfun 5265   -->wf 5267
This theorem is referenced by:  pwfi2f1o  27363
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-f 5275  df-fv 5279
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