Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ffs2 Structured version   Unicode version

Theorem ffs2 27984
Description: Rewrite a function's support based with its range rather than the universal class. See also frnsuppeq 6913. (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 6913 . . 3  |-  ( ( A  e.  V  /\  Z  e.  W )  ->  ( F : A --> B  ->  ( F supp  Z
)  =  ( `' F " ( B 
\  { Z }
) ) ) )
213impia 1194 . 2  |-  ( ( A  e.  V  /\  Z  e.  W  /\  F : A --> B )  ->  ( F supp  Z
)  =  ( `' F " ( B 
\  { Z }
) ) )
3 ffs2.1 . . 3  |-  C  =  ( B  \  { Z } )
43imaeq2i 5154 . 2  |-  ( `' F " C )  =  ( `' F " ( B  \  { Z } ) )
52, 4syl6eqr 2461 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    /\ w3a 974    = wceq 1405    e. wcel 1842    \ cdif 3410   {csn 3971   `'ccnv 4821   "cima 4825   -->wf 5564  (class class class)co 6277   supp csupp 6901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-supp 6902
This theorem is referenced by:  resf1o  27986  fsumcvg4  28371  eulerpartlems  28791  eulerpartlemgf  28810
  Copyright terms: Public domain W3C validator