MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ressuppfi Structured version   Visualization version   GIF version

Theorem ressuppfi 9392
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 2732 . . . . 5 (𝜑 → (𝐹𝐵) = 𝐺)
32oveq1d 7420 . . . 4 (𝜑 → ((𝐹𝐵) supp 𝑍) = (𝐺 supp 𝑍))
4 ressuppfi.s . . . 4 (𝜑 → (𝐺 supp 𝑍) ∈ Fin)
53, 4eqeltrd 2827 . . 3 (𝜑 → ((𝐹𝐵) supp 𝑍) ∈ Fin)
6 ressuppfi.b . . 3 (𝜑 → (dom 𝐹𝐵) ∈ Fin)
7 unfi 9174 . . 3 ((((𝐹𝐵) supp 𝑍) ∈ Fin ∧ (dom 𝐹𝐵) ∈ Fin) → (((𝐹𝐵) supp 𝑍) ∪ (dom 𝐹𝐵)) ∈ Fin)
85, 6, 7syl2anc 583 . 2 (𝜑 → (((𝐹𝐵) supp 𝑍) ∪ (dom 𝐹𝐵)) ∈ Fin)
9 ressuppfi.f . . 3 (𝜑𝐹𝑊)
10 ressuppfi.z . . 3 (𝜑𝑍𝑉)
11 ressuppssdif 8170 . . 3 ((𝐹𝑊𝑍𝑉) → (𝐹 supp 𝑍) ⊆ (((𝐹𝐵) supp 𝑍) ∪ (dom 𝐹𝐵)))
129, 10, 11syl2anc 583 . 2 (𝜑 → (𝐹 supp 𝑍) ⊆ (((𝐹𝐵) supp 𝑍) ∪ (dom 𝐹𝐵)))
138, 12ssfid 9269 1 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cdif 3940  cun 3941  wss 3943  dom cdm 5669  cres 5671  (class class class)co 7405   supp csupp 8146  Fincfn 8941
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-supp 8147  df-1o 8467  df-en 8942  df-fin 8945
This theorem is referenced by:  resfsupp  9393
  Copyright terms: Public domain W3C validator