Theorem dprdw 19613

Description: The property of being a finitely supported function in the family 𝑆. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)

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)

Assertion

Ref	Expression
dprdw	⊢ (𝜑 → (𝐹 ∈ 𝑊 ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥) ∧ 𝐹 finSupp 0 )))

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

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

Proof of Theorem dprdw

Step	Hyp	Ref	Expression
1		elex 3450	. . . . 5 ⊢ (𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) → 𝐹 ∈ V)
2	1	a1i 11	. . . 4 ⊢ (𝜑 → (𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) → 𝐹 ∈ V))
3		dprdff.1	. . . . . . 7 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
4		dprdff.2	. . . . . . 7 ⊢ (𝜑 → dom 𝑆 = 𝐼)
5	3, 4	dprddomcld 19604	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V)
6		fnex 7093	. . . . . . 7 ⊢ ((𝐹 Fn 𝐼 ∧ 𝐼 ∈ V) → 𝐹 ∈ V)
7	6	expcom 414	. . . . . 6 ⊢ (𝐼 ∈ V → (𝐹 Fn 𝐼 → 𝐹 ∈ V))
8	5, 7	syl 17	. . . . 5 ⊢ (𝜑 → (𝐹 Fn 𝐼 → 𝐹 ∈ V))
9	8	adantrd 492	. . . 4 ⊢ (𝜑 → ((𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥)) → 𝐹 ∈ V))
10		fveq2 6774	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → (𝑆‘𝑖) = (𝑆‘𝑥))
11	10	cbvixpv 8703	. . . . . . . 8 ⊢ X𝑖 ∈ 𝐼 (𝑆‘𝑖) = X𝑥 ∈ 𝐼 (𝑆‘𝑥)
12	11	eleq2i 2830	. . . . . . 7 ⊢ (𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ↔ 𝐹 ∈ X𝑥 ∈ 𝐼 (𝑆‘𝑥))
13		elixp2 8689	. . . . . . 7 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐼 (𝑆‘𝑥) ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥)))
14		3anass 1094	. . . . . . 7 ⊢ ((𝐹 ∈ V ∧ 𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥)) ↔ (𝐹 ∈ V ∧ (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥))))
15	12, 13, 14	3bitri 297	. . . . . 6 ⊢ (𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ↔ (𝐹 ∈ V ∧ (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥))))
16	15	baib 536	. . . . 5 ⊢ (𝐹 ∈ V → (𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥))))
17	16	a1i 11	. . . 4 ⊢ (𝜑 → (𝐹 ∈ V → (𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥)))))
18	2, 9, 17	pm5.21ndd 381	. . 3 ⊢ (𝜑 → (𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥))))
19	18	anbi1d 630	. 2 ⊢ (𝜑 → ((𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∧ 𝐹 finSupp 0 ) ↔ ((𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥)) ∧ 𝐹 finSupp 0 )))
20		breq1 5077	. . 3 ⊢ (ℎ = 𝐹 → (ℎ finSupp 0 ↔ 𝐹 finSupp 0 ))
21		dprdff.w	. . 3 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }
22	20, 21	elrab2 3627	. 2 ⊢ (𝐹 ∈ 𝑊 ↔ (𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∧ 𝐹 finSupp 0 ))
23		df-3an 1088	. 2 ⊢ ((𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥) ∧ 𝐹 finSupp 0 ) ↔ ((𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥)) ∧ 𝐹 finSupp 0 ))
24	19, 22, 23	3bitr4g 314	1 ⊢ (𝜑 → (𝐹 ∈ 𝑊 ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥) ∧ 𝐹 finSupp 0 )))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068  Vcvv 3432   class class class wbr 5074  dom cdm 5589   Fn wfn 6428  ‘cfv 6433  Xcixp 8685   finSupp cfsupp 9128   DProd cdprd 19596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-oprab 7279  df-mpo 7280  df-ixp 8686  df-dprd 19598

This theorem is referenced by:  dprdff  19615  dprdfcl  19616  dprdffsupp  19617  dprdsubg  19627