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

Theorem resfsupp 8287
Description: If the restriction of a function by a set which, subtracted from the domain of the function so that its difference is finitely supported, the function itself is finitely supported. (Contributed by AV, 27-May-2019.)
Hypotheses
Ref Expression
resfsupp.b (𝜑 → (dom 𝐹𝐵) ∈ Fin)
resfsupp.e (𝜑𝐹𝑊)
resfsupp.f (𝜑 → Fun 𝐹)
resfsupp.g (𝜑𝐺 = (𝐹𝐵))
resfsupp.s (𝜑𝐺 finSupp 𝑍)
resfsupp.z (𝜑𝑍𝑉)
Assertion
Ref Expression
resfsupp (𝜑𝐹 finSupp 𝑍)

Proof of Theorem resfsupp
StepHypRef Expression
1 resfsupp.b . . 3 (𝜑 → (dom 𝐹𝐵) ∈ Fin)
2 resfsupp.e . . 3 (𝜑𝐹𝑊)
3 resfsupp.g . . 3 (𝜑𝐺 = (𝐹𝐵))
4 resfsupp.s . . . 4 (𝜑𝐺 finSupp 𝑍)
54fsuppimpd 8267 . . 3 (𝜑 → (𝐺 supp 𝑍) ∈ Fin)
6 resfsupp.z . . 3 (𝜑𝑍𝑉)
71, 2, 3, 5, 6ressuppfi 8286 . 2 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
8 resfsupp.f . . 3 (𝜑 → Fun 𝐹)
9 funisfsupp 8265 . . 3 ((Fun 𝐹𝐹𝑊𝑍𝑉) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
108, 2, 6, 9syl3anc 1324 . 2 (𝜑 → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
117, 10mpbird 247 1 (𝜑𝐹 finSupp 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1481  wcel 1988  cdif 3564   class class class wbr 4644  dom cdm 5104  cres 5106  Fun wfun 5870  (class class class)co 6635   supp csupp 7280  Fincfn 7940   finSupp cfsupp 8260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-supp 7281  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-oadd 7549  df-er 7727  df-en 7941  df-fin 7944  df-fsupp 8261
This theorem is referenced by:  lincext2  42009
  Copyright terms: Public domain W3C validator