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Theorem ressuppfi 9011
Description: If the support of the restriction of a function by a set which, subtracted from the domain of the function so that its difference is finite, the support of the function itself is finite. (Contributed by AV, 22-Apr-2019.)
Hypotheses
Ref Expression
ressuppfi.b (𝜑 → (dom 𝐹𝐵) ∈ Fin)
ressuppfi.f (𝜑𝐹𝑊)
ressuppfi.g (𝜑𝐺 = (𝐹𝐵))
ressuppfi.s (𝜑 → (𝐺 supp 𝑍) ∈ Fin)
ressuppfi.z (𝜑𝑍𝑉)
Assertion
Ref Expression
ressuppfi (𝜑 → (𝐹 supp 𝑍) ∈ Fin)

Proof of Theorem ressuppfi
StepHypRef Expression
1 ressuppfi.g . . . . . 6 (𝜑𝐺 = (𝐹𝐵))
21eqcomd 2743 . . . . 5 (𝜑 → (𝐹𝐵) = 𝐺)
32oveq1d 7228 . . . 4 (𝜑 → ((𝐹𝐵) supp 𝑍) = (𝐺 supp 𝑍))
4 ressuppfi.s . . . 4 (𝜑 → (𝐺 supp 𝑍) ∈ Fin)
53, 4eqeltrd 2838 . . 3 (𝜑 → ((𝐹𝐵) supp 𝑍) ∈ Fin)
6 ressuppfi.b . . 3 (𝜑 → (dom 𝐹𝐵) ∈ Fin)
7 unfi 8850 . . 3 ((((𝐹𝐵) supp 𝑍) ∈ Fin ∧ (dom 𝐹𝐵) ∈ Fin) → (((𝐹𝐵) supp 𝑍) ∪ (dom 𝐹𝐵)) ∈ Fin)
85, 6, 7syl2anc 587 . 2 (𝜑 → (((𝐹𝐵) supp 𝑍) ∪ (dom 𝐹𝐵)) ∈ Fin)
9 ressuppfi.f . . 3 (𝜑𝐹𝑊)
10 ressuppfi.z . . 3 (𝜑𝑍𝑉)
11 ressuppssdif 7927 . . 3 ((𝐹𝑊𝑍𝑉) → (𝐹 supp 𝑍) ⊆ (((𝐹𝐵) supp 𝑍) ∪ (dom 𝐹𝐵)))
129, 10, 11syl2anc 587 . 2 (𝜑 → (𝐹 supp 𝑍) ⊆ (((𝐹𝐵) supp 𝑍) ∪ (dom 𝐹𝐵)))
138, 12ssfid 8898 1 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  cdif 3863  cun 3864  wss 3866  dom cdm 5551  cres 5553  (class class class)co 7213   supp csupp 7903  Fincfn 8626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-supp 7904  df-1o 8202  df-en 8627  df-fin 8630
This theorem is referenced by:  resfsupp  9012
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