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Theorem resfsupp 8836
 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 8816 . . 3 (𝜑 → (𝐺 supp 𝑍) ∈ Fin)
6 resfsupp.z . . 3 (𝜑𝑍𝑉)
71, 2, 3, 5, 6ressuppfi 8835 . 2 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
8 resfsupp.f . . 3 (𝜑 → Fun 𝐹)
9 funisfsupp 8814 . . 3 ((Fun 𝐹𝐹𝑊𝑍𝑉) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
108, 2, 6, 9syl3anc 1368 . 2 (𝜑 → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
117, 10mpbird 260 1 (𝜑𝐹 finSupp 𝑍)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2115   ∖ cdif 3907   class class class wbr 5039  dom cdm 5528   ↾ cres 5530  Fun wfun 6322  (class class class)co 7130   supp csupp 7805  Fincfn 8484   finSupp cfsupp 8809 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-supp 7806  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-oadd 8081  df-er 8264  df-en 8485  df-fin 8488  df-fsupp 8810 This theorem is referenced by:  lincext2  44655
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