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Theorem resfsupp 9390
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 9368 . . 3 (𝜑 → (𝐺 supp 𝑍) ∈ Fin)
6 resfsupp.z . . 3 (𝜑𝑍𝑉)
71, 2, 3, 5, 6ressuppfi 9389 . 2 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
8 resfsupp.f . . 3 (𝜑 → Fun 𝐹)
9 funisfsupp 9366 . . 3 ((Fun 𝐹𝐹𝑊𝑍𝑉) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
108, 2, 6, 9syl3anc 1371 . 2 (𝜑 → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
117, 10mpbird 256 1 (𝜑𝐹 finSupp 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2106  cdif 3945   class class class wbr 5148  dom cdm 5676  cres 5678  Fun wfun 6537  (class class class)co 7408   supp csupp 8145  Fincfn 8938   finSupp cfsupp 9360
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-supp 8146  df-1o 8465  df-en 8939  df-fin 8942  df-fsupp 9361
This theorem is referenced by:  lincext2  47126
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