Theorem resfsupp 9323

Description: If the restriction of a function by a set which, subtracted from the domain of the function so that its difference is finitely supported, the function itself is finitely supported. (Contributed by AV, 27-May-2019.)

Hypotheses

Ref	Expression
resfsupp.b	⊢ (𝜑 → (dom 𝐹 ∖ 𝐵) ∈ Fin)
resfsupp.e	⊢ (𝜑 → 𝐹 ∈ 𝑊)
resfsupp.f	⊢ (𝜑 → Fun 𝐹)
resfsupp.g	⊢ (𝜑 → 𝐺 = (𝐹 ↾ 𝐵))
resfsupp.s	⊢ (𝜑 → 𝐺 finSupp 𝑍)
resfsupp.z	⊢ (𝜑 → 𝑍 ∈ 𝑉)

Assertion

Ref	Expression
resfsupp	⊢ (𝜑 → 𝐹 finSupp 𝑍)

Proof of Theorem resfsupp

Step	Hyp	Ref	Expression
1		resfsupp.b	. . 3 ⊢ (𝜑 → (dom 𝐹 ∖ 𝐵) ∈ Fin)
2		resfsupp.e	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊)
3		resfsupp.g	. . 3 ⊢ (𝜑 → 𝐺 = (𝐹 ↾ 𝐵))
4		resfsupp.s	. . . 4 ⊢ (𝜑 → 𝐺 finSupp 𝑍)
5	4	fsuppimpd 9296	. . 3 ⊢ (𝜑 → (𝐺 supp 𝑍) ∈ Fin)
6		resfsupp.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
7	1, 2, 3, 5, 6	ressuppfi 9322	. 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
8		resfsupp.f	. . 3 ⊢ (𝜑 → Fun 𝐹)
9		funisfsupp 9294	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
10	8, 2, 6, 9	syl3anc 1373	. 2 ⊢ (𝜑 → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
11	7, 10	mpbird 257	1 ⊢ (𝜑 → 𝐹 finSupp 𝑍)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109   ∖ cdif 3908   class class class wbr 5102  dom cdm 5631   ↾ cres 5633  Fun wfun 6493  (class class class)co 7369   supp csupp 8116  Fincfn 8895   finSupp cfsupp 9288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-supp 8117  df-1o 8411  df-en 8896  df-fin 8899  df-fsupp 9289

This theorem is referenced by:  lincext2  48417