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Theorem ressuppfi 8861
 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 2804 . . . . 5 (𝜑 → (𝐹𝐵) = 𝐺)
32oveq1d 7160 . . . 4 (𝜑 → ((𝐹𝐵) supp 𝑍) = (𝐺 supp 𝑍))
4 ressuppfi.s . . . 4 (𝜑 → (𝐺 supp 𝑍) ∈ Fin)
53, 4eqeltrd 2890 . . 3 (𝜑 → ((𝐹𝐵) supp 𝑍) ∈ Fin)
6 ressuppfi.b . . 3 (𝜑 → (dom 𝐹𝐵) ∈ Fin)
7 unfi 8787 . . 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 7850 . . 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 8743 1 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111   ∖ cdif 3880   ∪ cun 3881   ⊆ wss 3883  dom cdm 5523   ↾ cres 5525  (class class class)co 7145   supp csupp 7826  Fincfn 8510 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-supp 7827  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-oadd 8107  df-er 8290  df-en 8511  df-fin 8514 This theorem is referenced by:  resfsupp  8862
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