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Theorem fsuppsssuppgd 41534
Description: If the support of a function is a subset of a finite support, it is finite. Deduction associated with fsuppsssupp 9385. (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 9375 . 2 (𝜑 → (𝐹 supp 𝑂) ∈ Fin)
6 fsuppsssuppgd.3 . 2 (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑂))
7 suppssfifsupp 9384 . 2 (((𝐺𝑉 ∧ Fun 𝐺𝑍𝑊) ∧ ((𝐹 supp 𝑂) ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑂))) → 𝐺 finSupp 𝑍)
81, 2, 3, 5, 6, 7syl32anc 1377 1 (𝜑𝐺 finSupp 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  wss 3948   class class class wbr 5148  Fun wfun 6537  (class class class)co 7412   supp csupp 8151  Fincfn 8945   finSupp cfsupp 9367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  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 7415  df-om 7860  df-1o 8472  df-en 8946  df-fin 8949  df-fsupp 9368
This theorem is referenced by:  fsuppss  41535  evlsvvvallem  41599  evlsvvvallem2  41600  evlsvvval  41601  selvvvval  41623  evlselv  41625  mhphf  41635
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