Theorem fsuppsssuppgd 9299

Description: If the support of a function is a subset of a finite support, it is finite. Deduction associated with fsuppsssupp 9298. (Contributed by SN, 6-Mar-2025.)

Hypotheses

Ref	Expression
fsuppsssuppgd.g	⊢ (𝜑 → 𝐺 ∈ 𝑉)
fsuppsssuppgd.z	⊢ (𝜑 → 𝑍 ∈ 𝑊)
fsuppsssuppgd.1	⊢ (𝜑 → Fun 𝐺)
fsuppsssuppgd.2	⊢ (𝜑 → 𝐹 finSupp 𝑂)
fsuppsssuppgd.3	⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑂))

Assertion

Ref	Expression
fsuppsssuppgd	⊢ (𝜑 → 𝐺 finSupp 𝑍)

Proof of Theorem fsuppsssuppgd

Step	Hyp	Ref	Expression
1		fsuppsssuppgd.g	. 2 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
2		fsuppsssuppgd.1	. 2 ⊢ (𝜑 → Fun 𝐺)
3		fsuppsssuppgd.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
4		fsuppsssuppgd.2	. . 3 ⊢ (𝜑 → 𝐹 finSupp 𝑂)
5	4	fsuppimpd 9286	. 2 ⊢ (𝜑 → (𝐹 supp 𝑂) ∈ Fin)
6		fsuppsssuppgd.3	. 2 ⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑂))
7		suppssfifsupp 9297	. 2 ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺 ∧ 𝑍 ∈ 𝑊) ∧ ((𝐹 supp 𝑂) ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑂))) → 𝐺 finSupp 𝑍)
8	1, 2, 3, 5, 6, 7	syl32anc 1381	1 ⊢ (𝜑 → 𝐺 finSupp 𝑍)

Syntax hints:   → wi 4   ∈ wcel 2114   ⊆ wss 3903   class class class wbr 5100  Fun wfun 6496  (class class class)co 7370   supp csupp 8114  Fincfn 8897   finSupp cfsupp 9278

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ov 7373  df-om 7821  df-1o 8409  df-en 8898  df-fin 8901  df-fsupp 9279

This theorem is referenced by:  fsuppss  9300  fsuppssov1  9301  evlsvvvallem  22063  evlsvvvallem2  22064  evlsvvval  22065  fisuppov1  32779  elrgspnlem1  33342  mplvrpmrhm  33730  fldextrspunlsp  33858  selvvvval  42972  evlselv  42974  mhphf  42984