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Theorem fsuppsssuppgd 9400
Description: If the support of a function is a subset of a finite support, it is finite. Deduction associated with fsuppsssupp 9399. (Contributed by SN, 6-Mar-2025.)
Hypotheses
Ref Expression
fsuppsssuppgd.g (𝜑𝐺𝑉)
fsuppsssuppgd.z (𝜑𝑍𝑊)
fsuppsssuppgd.1 (𝜑 → Fun 𝐺)
fsuppsssuppgd.2 (𝜑𝐹 finSupp 𝑂)
fsuppsssuppgd.3 (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑂))
Assertion
Ref Expression
fsuppsssuppgd (𝜑𝐺 finSupp 𝑍)

Proof of Theorem fsuppsssuppgd
StepHypRef Expression
1 fsuppsssuppgd.g . 2 (𝜑𝐺𝑉)
2 fsuppsssuppgd.1 . 2 (𝜑 → Fun 𝐺)
3 fsuppsssuppgd.z . 2 (𝜑𝑍𝑊)
4 fsuppsssuppgd.2 . . 3 (𝜑𝐹 finSupp 𝑂)
54fsuppimpd 9388 . 2 (𝜑 → (𝐹 supp 𝑂) ∈ Fin)
6 fsuppsssuppgd.3 . 2 (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑂))
7 suppssfifsupp 9398 . 2 (((𝐺𝑉 ∧ Fun 𝐺𝑍𝑊) ∧ ((𝐹 supp 𝑂) ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑂))) → 𝐺 finSupp 𝑍)
81, 2, 3, 5, 6, 7syl32anc 1375 1 (𝜑𝐺 finSupp 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wss 3941   class class class wbr 5144  Fun wfun 6537  (class class class)co 7413   supp csupp 8158  Fincfn 8957   finSupp cfsupp 9380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  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 7416  df-om 7866  df-1o 8480  df-en 8958  df-fin 8961  df-fsupp 9381
This theorem is referenced by:  fsuppss  9401  fsuppssov1  9402  evlsvvvallem  41855  evlsvvvallem2  41856  evlsvvval  41857  selvvvval  41879  evlselv  41881  mhphf  41891
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