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Theorem fsuppeq 26921
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 3518 . . 3  |-  ( S  i^i  ( _V  \  { X } ) )  =  ( S  \  { X } )
21imaeq2i 5134 . 2  |-  ( `' F " ( S  i^i  ( _V  \  { X } ) ) )  =  ( `' F " ( S 
\  { X }
) )
3 ffun 5526 . . . 4  |-  ( F : I --> S  ->  Fun  F )
4 inpreima 5789 . . . 4  |-  ( Fun 
F  ->  ( `' F " ( S  i^i  ( _V  \  { X } ) ) )  =  ( ( `' F " S )  i^i  ( `' F " ( _V  \  { X } ) ) ) )
53, 4syl 16 . . 3  |-  ( F : I --> S  -> 
( `' F "
( S  i^i  ( _V  \  { X }
) ) )  =  ( ( `' F " S )  i^i  ( `' F " ( _V 
\  { X }
) ) ) )
6 cnvimass 5157 . . . . 5  |-  ( `' F " ( _V 
\  { X }
) )  C_  dom  F
7 fdm 5528 . . . . . 6  |-  ( F : I --> S  ->  dom  F  =  I )
8 fimacnv 5794 . . . . . 6  |-  ( F : I --> S  -> 
( `' F " S )  =  I )
97, 8eqtr4d 2415 . . . . 5  |-  ( F : I --> S  ->  dom  F  =  ( `' F " S ) )
106, 9syl5sseq 3332 . . . 4  |-  ( F : I --> S  -> 
( `' F "
( _V  \  { X } ) )  C_  ( `' F " S ) )
11 sseqin2 3496 . . . 4  |-  ( ( `' F " ( _V 
\  { X }
) )  C_  ( `' F " S )  <-> 
( ( `' F " S )  i^i  ( `' F " ( _V 
\  { X }
) ) )  =  ( `' F "
( _V  \  { X } ) ) )
1210, 11sylib 189 . . 3  |-  ( F : I --> S  -> 
( ( `' F " S )  i^i  ( `' F " ( _V 
\  { X }
) ) )  =  ( `' F "
( _V  \  { X } ) ) )
135, 12eqtrd 2412 . 2  |-  ( F : I --> S  -> 
( `' F "
( S  i^i  ( _V  \  { X }
) ) )  =  ( `' F "
( _V  \  { X } ) ) )
142, 13syl5reqr 2427 1  |-  ( F : I --> S  -> 
( `' F "
( _V  \  { X } ) )  =  ( `' F "
( S  \  { X } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   _Vcvv 2892    \ cdif 3253    i^i cin 3255    C_ wss 3256   {csn 3750   `'ccnv 4810   dom cdm 4811   "cima 4814   Fun wfun 5381   -->wf 5383
This theorem is referenced by:  pwfi2f1o  26922
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395
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