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Theorem fsuppsssuppgd 41641
Description: If the support of a function is a subset of a finite support, it is finite. Deduction associated with fsuppsssupp 9393. (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 9383 . 2 (𝜑 → (𝐹 supp 𝑂) ∈ Fin)
6 fsuppsssuppgd.3 . 2 (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑂))
7 suppssfifsupp 9392 . 2 (((𝐺𝑉 ∧ Fun 𝐺𝑍𝑊) ∧ ((𝐹 supp 𝑂) ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑂))) → 𝐺 finSupp 𝑍)
81, 2, 3, 5, 6, 7syl32anc 1376 1 (𝜑𝐺 finSupp 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  wss 3944   class class class wbr 5142  Fun wfun 6536  (class class class)co 7414   supp csupp 8157  Fincfn 8953   finSupp cfsupp 9375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-om 7863  df-1o 8478  df-en 8954  df-fin 8957  df-fsupp 9376
This theorem is referenced by:  fsuppss  41642  evlsvvvallem  41706  evlsvvvallem2  41707  evlsvvval  41708  selvvvval  41730  evlselv  41732  mhphf  41742
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