resfifsupp - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > resfifsupp		Structured version Visualization version GIF version

Theorem resfifsupp 9337

Description: The restriction of a function to a finite set is finitely supported. (Contributed by AV, 12-Dec-2019.)

**Hypotheses**
Ref	Expression
resfifsupp.f	⊢ (𝜑 → Fun 𝐹)
resfifsupp.x	⊢ (𝜑 → 𝑋 ∈ Fin)
resfifsupp.z	⊢ (𝜑 → 𝑍 ∈ 𝑉)

**Assertion**
Ref	Expression
resfifsupp	⊢ (𝜑 → (𝐹 ↾ 𝑋) finSupp 𝑍)

**Proof of Theorem resfifsupp**
Step	Hyp	Ref	Expression
1		resindm 6012	. 2 ⊢ (𝐹 ↾ (𝑋 ∩ dom 𝐹)) = (𝐹 ↾ 𝑋)
2		resfifsupp.f	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
3	2	funfnd 6547	. . . 4 ⊢ (𝜑 → 𝐹 Fn dom 𝐹)
4		fnresin2 6642	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝐹 ↾ (𝑋 ∩ dom 𝐹)) Fn (𝑋 ∩ dom 𝐹))
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝑋 ∩ dom 𝐹)) Fn (𝑋 ∩ dom 𝐹))
6		resfifsupp.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin)
7		infi 9208	. . . 4 ⊢ (𝑋 ∈ Fin → (𝑋 ∩ dom 𝐹) ∈ Fin)
8	6, 7	syl 17	. . 3 ⊢ (𝜑 → (𝑋 ∩ dom 𝐹) ∈ Fin)
9		resfifsupp.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
10	5, 8, 9	fndmfifsupp 9318	. 2 ⊢ (𝜑 → (𝐹 ↾ (𝑋 ∩ dom 𝐹)) finSupp 𝑍)
11	1, 10	eqbrtrrid 5133	1 ⊢ (𝜑 → (𝐹 ↾ 𝑋) finSupp 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2141 ∩ cin 3901 class class class wbr 5097 dom cdm 5643 ↾ cres 5645 Fun wfun 6510 Fn wfn 6511 Fincfn 8921 finSupp cfsupp 9301

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pr 5387 ax-un 7713

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-supp 8135 df-1o 8431 df-en 8922 df-fin 8925 df-fsupp 9302

This theorem is referenced by: xrge0tsmsd 33214 ply1degltdimlem 33880

W3C validator