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

Theorem ffs2 32531
Description: Rewrite a function's support based with its codomain rather than the universal class. See also fsuppeq 8186. (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 ((𝐴𝑉𝑍𝑊𝐹:𝐴𝐵) → (𝐹 supp 𝑍) = (𝐹𝐶))

Proof of Theorem ffs2
StepHypRef Expression
1 fsuppeq 8186 . . 3 ((𝐴𝑉𝑍𝑊) → (𝐹:𝐴𝐵 → (𝐹 supp 𝑍) = (𝐹 “ (𝐵 ∖ {𝑍}))))
213impia 1114 . 2 ((𝐴𝑉𝑍𝑊𝐹:𝐴𝐵) → (𝐹 supp 𝑍) = (𝐹 “ (𝐵 ∖ {𝑍})))
3 ffs2.1 . . 3 𝐶 = (𝐵 ∖ {𝑍})
43imaeq2i 6066 . 2 (𝐹𝐶) = (𝐹 “ (𝐵 ∖ {𝑍}))
52, 4eqtr4di 2786 1 ((𝐴𝑉𝑍𝑊𝐹:𝐴𝐵) → (𝐹 supp 𝑍) = (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098  cdif 3946  {csn 4632  ccnv 5681  cima 5685  wf 6549  (class class class)co 7426   supp csupp 8171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-supp 8172
This theorem is referenced by:  resf1o  32533  fsumcvg4  33584  eulerpartlems  34013  eulerpartlemgf  34032
  Copyright terms: Public domain W3C validator