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Theorem ffs2 27220
Description: Rewrite a function's support based with its range rather than the universal class. See also frnsuppeq 6910. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
Hypothesis
Ref Expression
ffs2.1  |-  C  =  ( B  \  { Z } )
Assertion
Ref Expression
ffs2  |-  ( ( A  e.  V  /\  Z  e.  W  /\  F : A --> B )  ->  ( F supp  Z
)  =  ( `' F " C ) )

Proof of Theorem ffs2
StepHypRef Expression
1 frnsuppeq 6910 . . . 4  |-  ( ( A  e.  V  /\  Z  e.  W )  ->  ( F : A --> B  ->  ( F supp  Z
)  =  ( `' F " ( B 
\  { Z }
) ) ) )
21imp 429 . . 3  |-  ( ( ( A  e.  V  /\  Z  e.  W
)  /\  F : A
--> B )  ->  ( F supp  Z )  =  ( `' F " ( B 
\  { Z }
) ) )
323impa 1191 . 2  |-  ( ( A  e.  V  /\  Z  e.  W  /\  F : A --> B )  ->  ( F supp  Z
)  =  ( `' F " ( B 
\  { Z }
) ) )
4 ffs2.1 . . 3  |-  C  =  ( B  \  { Z } )
54imaeq2i 5333 . 2  |-  ( `' F " C )  =  ( `' F " ( B  \  { Z } ) )
63, 5syl6eqr 2526 1  |-  ( ( A  e.  V  /\  Z  e.  W  /\  F : A --> B )  ->  ( F supp  Z
)  =  ( `' F " C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    \ cdif 3473   {csn 4027   `'ccnv 4998   "cima 5002   -->wf 5582  (class class class)co 6282   supp csupp 6898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-supp 6899
This theorem is referenced by:  resf1o  27222  fsumcvg4  27565  eulerpartlems  27936  eulerpartlemgf  27955
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