Theorem dprdwd 19980

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 5076	. . 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 3131	. . . . 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 19970	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V)
7		mptelixpg 8874	. . . . . 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 258	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ X𝑥 ∈ 𝐼 (𝑆‘𝑥))
10		fveq2 6828	. . . . 5 ⊢ (𝑥 = 𝑖 → (𝑆‘𝑥) = (𝑆‘𝑖))
11	10	cbvixpv 8854	. . . 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 3631	. 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 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  {crab 3391  Vcvv 3431   class class class wbr 5073   ↦ cmpt 5154  dom cdm 5619  ‘cfv 6486  Xcixp 8836   finSupp cfsupp 9265   DProd cdprd 19962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-oprab 7361  df-mpo 7362  df-ixp 8837  df-dprd 19964

This theorem is referenced by:  dprdfid  19986  dprdfinv  19988  dprdfadd  19989  dmdprdsplitlem  20006  dpjidcl  20027  dchrptlem3  27248