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Theorem ffs2 32494
Description: Rewrite a function's support based with its codomain rather than the universal class. See also fsuppeq 8173. (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 8173 . . 3 ((𝐴𝑉𝑍𝑊) → (𝐹:𝐴𝐵 → (𝐹 supp 𝑍) = (𝐹 “ (𝐵 ∖ {𝑍}))))
213impia 1115 . 2 ((𝐴𝑉𝑍𝑊𝐹:𝐴𝐵) → (𝐹 supp 𝑍) = (𝐹 “ (𝐵 ∖ {𝑍})))
3 ffs2.1 . . 3 𝐶 = (𝐵 ∖ {𝑍})
43imaeq2i 6055 . 2 (𝐹𝐶) = (𝐹 “ (𝐵 ∖ {𝑍}))
52, 4eqtr4di 2785 1 ((𝐴𝑉𝑍𝑊𝐹:𝐴𝐵) → (𝐹 supp 𝑍) = (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1085   = wceq 1534  wcel 2099  cdif 3941  {csn 4624  ccnv 5671  cima 5675  wf 6538  (class class class)co 7414   supp csupp 8159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-supp 8160
This theorem is referenced by:  resf1o  32496  fsumcvg4  33487  eulerpartlems  33916  eulerpartlemgf  33935
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