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Theorem ffs2 26033
Description: Rewrite a function's support based with its range rather than the universal class. See also frnsuppeq 6707. (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 6707 . . . 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 1182 . 2  |-  ( ( A  e.  V  /\  Z  e.  W  /\  F : A --> B )  ->  ( F supp  Z
)  =  ( `' F " ( B 
\  { Z }
) ) )
4 ffs2.1 . . 3  |-  C  =  ( B  \  { Z } )
54imaeq2i 5172 . 2  |-  ( `' F " C )  =  ( `' F " ( B  \  { Z } ) )
63, 5syl6eqr 2493 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 965    = wceq 1369    e. wcel 1756    \ cdif 3330   {csn 3882   `'ccnv 4844   "cima 4848   -->wf 5419  (class class class)co 6096   supp csupp 6695
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pr 4536  ax-un 6377
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-supp 6696
This theorem is referenced by:  resf1o  26035  fsumcvg4  26385  eulerpartlems  26748  eulerpartlemgf  26767
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