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Theorem resfsupp 8185
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 8165 . . 3 (𝜑 → (𝐺 supp 𝑍) ∈ Fin)
6 resfsupp.z . . 3 (𝜑𝑍𝑉)
71, 2, 3, 5, 6ressuppfi 8184 . 2 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
8 resfsupp.f . . 3 (𝜑 → Fun 𝐹)
9 funisfsupp 8163 . . 3 ((Fun 𝐹𝐹𝑊𝑍𝑉) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
108, 2, 6, 9syl3anc 1318 . 2 (𝜑 → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
117, 10mpbird 246 1 (𝜑𝐹 finSupp 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195   = wceq 1475  wcel 1977  cdif 3537   class class class wbr 4583  dom cdm 5038  cres 5040  Fun wfun 5798  (class class class)co 6549   supp csupp 7182  Fincfn 7841   finSupp cfsupp 8158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fsupp 8159
This theorem is referenced by:  lincext2  42038
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