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Theorem resfifsupp 8935
Description: The restriction of a function to a finite set is finitely supported. (Contributed by AV, 12-Dec-2019.)
Hypotheses
Ref Expression
resfifsupp.f (𝜑 → Fun 𝐹)
resfifsupp.x (𝜑𝑋 ∈ Fin)
resfifsupp.z (𝜑𝑍𝑉)
Assertion
Ref Expression
resfifsupp (𝜑 → (𝐹𝑋) finSupp 𝑍)

Proof of Theorem resfifsupp
StepHypRef Expression
1 resfifsupp.f . . . 4 (𝜑 → Fun 𝐹)
2 funrel 6357 . . . 4 (Fun 𝐹 → Rel 𝐹)
31, 2syl 17 . . 3 (𝜑 → Rel 𝐹)
4 resindm 5875 . . 3 (Rel 𝐹 → (𝐹 ↾ (𝑋 ∩ dom 𝐹)) = (𝐹𝑋))
53, 4syl 17 . 2 (𝜑 → (𝐹 ↾ (𝑋 ∩ dom 𝐹)) = (𝐹𝑋))
61funfnd 6371 . . . 4 (𝜑𝐹 Fn dom 𝐹)
7 fnresin2 6463 . . . 4 (𝐹 Fn dom 𝐹 → (𝐹 ↾ (𝑋 ∩ dom 𝐹)) Fn (𝑋 ∩ dom 𝐹))
86, 7syl 17 . . 3 (𝜑 → (𝐹 ↾ (𝑋 ∩ dom 𝐹)) Fn (𝑋 ∩ dom 𝐹))
9 resfifsupp.x . . . 4 (𝜑𝑋 ∈ Fin)
10 infi 8821 . . . 4 (𝑋 ∈ Fin → (𝑋 ∩ dom 𝐹) ∈ Fin)
119, 10syl 17 . . 3 (𝜑 → (𝑋 ∩ dom 𝐹) ∈ Fin)
12 resfifsupp.z . . 3 (𝜑𝑍𝑉)
138, 11, 12fndmfifsupp 8920 . 2 (𝜑 → (𝐹 ↾ (𝑋 ∩ dom 𝐹)) finSupp 𝑍)
145, 13eqbrtrrd 5055 1 (𝜑 → (𝐹𝑋) finSupp 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2113  cin 3843   class class class wbr 5031  dom cdm 5526  cres 5528  Rel wrel 5531  Fun wfun 6334   Fn wfn 6335  Fincfn 8556   finSupp cfsupp 8907
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 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pr 5297  ax-un 7480
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3683  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7174  df-oprab 7175  df-mpo 7176  df-om 7601  df-supp 7858  df-1o 8132  df-en 8557  df-fin 8560  df-fsupp 8908
This theorem is referenced by:  xrge0tsmsd  30894
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