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Theorem fsuppeq 29403
Description: Two ways of writing the support of a function with known codomain. MOVABLE SHORTEN nn0supp (Contributed by Stefan O'Rear, 9-Jul-2015.) (New usage is discouraged.) Use frnsuppeq 6697 instead.
Assertion
Ref Expression
fsuppeq  |-  ( F : I --> S  -> 
( `' F "
( _V  \  { X } ) )  =  ( `' F "
( S  \  { X } ) ) )

Proof of Theorem fsuppeq
StepHypRef Expression
1 invdif 3586 . . 3  |-  ( S  i^i  ( _V  \  { X } ) )  =  ( S  \  { X } )
21imaeq2i 5162 . 2  |-  ( `' F " ( S  i^i  ( _V  \  { X } ) ) )  =  ( `' F " ( S 
\  { X }
) )
3 ffun 5556 . . . 4  |-  ( F : I --> S  ->  Fun  F )
4 inpreima 5825 . . . 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 5184 . . . . 5  |-  ( `' F " ( _V 
\  { X }
) )  C_  dom  F
7 fdm 5558 . . . . . 6  |-  ( F : I --> S  ->  dom  F  =  I )
8 fimacnv 5830 . . . . . 6  |-  ( F : I --> S  -> 
( `' F " S )  =  I )
97, 8eqtr4d 2473 . . . . 5  |-  ( F : I --> S  ->  dom  F  =  ( `' F " S ) )
106, 9syl5sseq 3399 . . . 4  |-  ( F : I --> S  -> 
( `' F "
( _V  \  { X } ) )  C_  ( `' F " S ) )
11 sseqin2 3564 . . . 4  |-  ( ( `' F " ( _V 
\  { X }
) )  C_  ( `' F " S )  <-> 
( ( `' F " S )  i^i  ( `' F " ( _V 
\  { X }
) ) )  =  ( `' F "
( _V  \  { X } ) ) )
1210, 11sylib 196 . . 3  |-  ( F : I --> S  -> 
( ( `' F " S )  i^i  ( `' F " ( _V 
\  { X }
) ) )  =  ( `' F "
( _V  \  { X } ) ) )
135, 12eqtrd 2470 . 2  |-  ( F : I --> S  -> 
( `' F "
( S  i^i  ( _V  \  { X }
) ) )  =  ( `' F "
( _V  \  { X } ) ) )
142, 13syl5reqr 2485 1  |-  ( F : I --> S  -> 
( `' F "
( _V  \  { X } ) )  =  ( `' F "
( S  \  { X } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369   _Vcvv 2967    \ cdif 3320    i^i cin 3322    C_ wss 3323   {csn 3872   `'ccnv 4834   dom cdm 4835   "cima 4838   Fun wfun 5407   -->wf 5409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421
This theorem is referenced by:  pwfi2f1o  29404
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