Theorem resfsupp 9421

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 9395	. . 3 ⊢ (𝜑 → (𝐺 supp 𝑍) ∈ Fin)
6		resfsupp.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
7	1, 2, 3, 5, 6	ressuppfi 9420	. 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
8		resfsupp.f	. . 3 ⊢ (𝜑 → Fun 𝐹)
9		funisfsupp 9393	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
10	8, 2, 6, 9	syl3anc 1368	. 2 ⊢ (𝜑 → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
11	7, 10	mpbird 256	1 ⊢ (𝜑 → 𝐹 finSupp 𝑍)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098   ∖ cdif 3941   class class class wbr 5149  dom cdm 5678   ↾ cres 5680  Fun wfun 6543  (class class class)co 7419   supp csupp 8165  Fincfn 8964   finSupp cfsupp 9387

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 5300  ax-nul 5307  ax-pr 5429  ax-un 7741

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 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-supp 8166  df-1o 8487  df-en 8965  df-fin 8968  df-fsupp 9388

This theorem is referenced by:  lincext2  47709