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Theorem fsuppeq 27259
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 3410 . . 3  |-  ( S  i^i  ( _V  \  { X } ) )  =  ( S  \  { X } )
21imaeq2i 5010 . 2  |-  ( `' F " ( S  i^i  ( _V  \  { X } ) ) )  =  ( `' F " ( S 
\  { X }
) )
3 ffun 5391 . . . 4  |-  ( F : I --> S  ->  Fun  F )
4 inpreima 5652 . . . 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 5033 . . . . 5  |-  ( `' F " ( _V 
\  { X }
) )  C_  dom  F
7 fdm 5393 . . . . . 6  |-  ( F : I --> S  ->  dom  F  =  I )
8 fimacnv 5657 . . . . . 6  |-  ( F : I --> S  -> 
( `' F " S )  =  I )
97, 8eqtr4d 2318 . . . . 5  |-  ( F : I --> S  ->  dom  F  =  ( `' F " S ) )
106, 9syl5sseq 3226 . . . 4  |-  ( F : I --> S  -> 
( `' F "
( _V  \  { X } ) )  C_  ( `' F " S ) )
11 sseqin2 3388 . . . 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 2315 . 2  |-  ( F : I --> S  -> 
( `' F "
( S  i^i  ( _V  \  { X }
) ) )  =  ( `' F "
( _V  \  { X } ) ) )
142, 13syl5reqr 2330 1  |-  ( F : I --> S  -> 
( `' F "
( _V  \  { X } ) )  =  ( `' F "
( S  \  { X } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   {csn 3640   `'ccnv 4688   dom cdm 4689   "cima 4692   Fun wfun 5249   -->wf 5251
This theorem is referenced by:  pwfi2f1o  27260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263
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