Theorem dprdwd 19925

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 5092	. . 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 3124	. . . . 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 19915	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V)
7		mptelixpg 8859	. . . . . 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 257	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ X𝑥 ∈ 𝐼 (𝑆‘𝑥))
10		fveq2 6822	. . . . 5 ⊢ (𝑥 = 𝑖 → (𝑆‘𝑥) = (𝑆‘𝑖))
11	10	cbvixpv 8839	. . . 4 ⊢ X𝑥 ∈ 𝐼 (𝑆‘𝑥) = X𝑖 ∈ 𝐼 (𝑆‘𝑖)
12	9, 11	eleqtrdi 2841	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖))
13		dprdwd.4	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) finSupp 0 )
14	1, 12, 13	elrabd 3644	. 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 2842	1 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ 𝑊)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  {crab 3395  Vcvv 3436   class class class wbr 5089   ↦ cmpt 5170  dom cdm 5614  ‘cfv 6481  Xcixp 8821   finSupp cfsupp 9245   DProd cdprd 19907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-oprab 7350  df-mpo 7351  df-ixp 8822  df-dprd 19909

This theorem is referenced by:  dprdfid  19931  dprdfinv  19933  dprdfadd  19934  dmdprdsplitlem  19951  dpjidcl  19972  dchrptlem3  27204