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Theorem resfsupp 9133
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 9113 . . 3 (𝜑 → (𝐺 supp 𝑍) ∈ Fin)
6 resfsupp.z . . 3 (𝜑𝑍𝑉)
71, 2, 3, 5, 6ressuppfi 9132 . 2 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
8 resfsupp.f . . 3 (𝜑 → Fun 𝐹)
9 funisfsupp 9111 . . 3 ((Fun 𝐹𝐹𝑊𝑍𝑉) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
108, 2, 6, 9syl3anc 1370 . 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 1542  wcel 2110  cdif 3889   class class class wbr 5079  dom cdm 5590  cres 5592  Fun wfun 6426  (class class class)co 7271   supp csupp 7968  Fincfn 8716   finSupp cfsupp 9106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-supp 7969  df-1o 8288  df-en 8717  df-fin 8720  df-fsupp 9107
This theorem is referenced by:  lincext2  45765
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