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

Theorem ressuppfi 9390
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 2739 . . . . 5 (𝜑 → (𝐹𝐵) = 𝐺)
32oveq1d 7424 . . . 4 (𝜑 → ((𝐹𝐵) supp 𝑍) = (𝐺 supp 𝑍))
4 ressuppfi.s . . . 4 (𝜑 → (𝐺 supp 𝑍) ∈ Fin)
53, 4eqeltrd 2834 . . 3 (𝜑 → ((𝐹𝐵) supp 𝑍) ∈ Fin)
6 ressuppfi.b . . 3 (𝜑 → (dom 𝐹𝐵) ∈ Fin)
7 unfi 9172 . . 3 ((((𝐹𝐵) supp 𝑍) ∈ Fin ∧ (dom 𝐹𝐵) ∈ Fin) → (((𝐹𝐵) supp 𝑍) ∪ (dom 𝐹𝐵)) ∈ Fin)
85, 6, 7syl2anc 585 . 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 585 . 2 (𝜑 → (𝐹 supp 𝑍) ⊆ (((𝐹𝐵) supp 𝑍) ∪ (dom 𝐹𝐵)))
138, 12ssfid 9267 1 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cdif 3946  cun 3947  wss 3949  dom cdm 5677  cres 5679  (class class class)co 7409   supp csupp 8146  Fincfn 8939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-supp 8147  df-1o 8466  df-en 8940  df-fin 8943
This theorem is referenced by:  resfsupp  9391
  Copyright terms: Public domain W3C validator