Theorem dprdwd 19529

Description: A mapping being a finitely supported function in the family 𝑆. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.) (Proof shortened by OpenAI, 30-Mar-2020.)

Hypotheses

Ref	Expression
dprdff.w	⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }
dprdff.1	⊢ (𝜑 → 𝐺dom DProd 𝑆)
dprdff.2	⊢ (𝜑 → dom 𝑆 = 𝐼)
dprdwd.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ (𝑆‘𝑥))
dprdwd.4	⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) finSupp 0 )

Assertion

Ref	Expression
dprdwd	⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ 𝑊)

Distinct variable groups:   𝐴,ℎ   𝑥,ℎ   𝑥,𝐺   ℎ,𝑖,𝐼,𝑥   0 ,ℎ   𝜑,𝑥   𝑆,ℎ,𝑖,𝑥

Allowed substitution hints:   𝜑(ℎ,𝑖)   𝐴(𝑥,𝑖)   𝐺(ℎ,𝑖)   𝑊(𝑥,ℎ,𝑖)   0 (𝑥,𝑖)

Proof of Theorem dprdwd

Step	Hyp	Ref	Expression
1		breq1 5073	. . 3 ⊢ (ℎ = (𝑥 ∈ 𝐼 ↦ 𝐴) → (ℎ finSupp 0 ↔ (𝑥 ∈ 𝐼 ↦ 𝐴) finSupp 0 ))
2		dprdwd.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ (𝑆‘𝑥))
3	2	ralrimiva 3107	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝐴 ∈ (𝑆‘𝑥))
4		dprdff.1	. . . . . . 7 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
5		dprdff.2	. . . . . . 7 ⊢ (𝜑 → dom 𝑆 = 𝐼)
6	4, 5	dprddomcld 19519	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V)
7		mptelixpg 8681	. . . . . 6 ⊢ (𝐼 ∈ V → ((𝑥 ∈ 𝐼 ↦ 𝐴) ∈ X𝑥 ∈ 𝐼 (𝑆‘𝑥) ↔ ∀𝑥 ∈ 𝐼 𝐴 ∈ (𝑆‘𝑥)))
8	6, 7	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ 𝐴) ∈ X𝑥 ∈ 𝐼 (𝑆‘𝑥) ↔ ∀𝑥 ∈ 𝐼 𝐴 ∈ (𝑆‘𝑥)))
9	3, 8	mpbird 256	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ X𝑥 ∈ 𝐼 (𝑆‘𝑥))
10		fveq2 6756	. . . . 5 ⊢ (𝑥 = 𝑖 → (𝑆‘𝑥) = (𝑆‘𝑖))
11	10	cbvixpv 8661	. . . 4 ⊢ X𝑥 ∈ 𝐼 (𝑆‘𝑥) = X𝑖 ∈ 𝐼 (𝑆‘𝑖)
12	9, 11	eleqtrdi 2849	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖))
13		dprdwd.4	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) finSupp 0 )
14	1, 12, 13	elrabd 3619	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 })
15		dprdff.w	. 2 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }
16	14, 15	eleqtrrdi 2850	1 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ 𝑊)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067  Vcvv 3422   class class class wbr 5070   ↦ cmpt 5153  dom cdm 5580  ‘cfv 6418  Xcixp 8643   finSupp cfsupp 9058   DProd cdprd 19511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-oprab 7259  df-mpo 7260  df-ixp 8644  df-dprd 19513

This theorem is referenced by:  dprdfid  19535  dprdfinv  19537  dprdfadd  19538  dmdprdsplitlem  19555  dpjidcl  19576  dchrptlem3  26319