Theorem dprdw 20045

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 3499	. . . . 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 20036	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V)
6		fnex 7237	. . . . . . 7 ⊢ ((𝐹 Fn 𝐼 ∧ 𝐼 ∈ V) → 𝐹 ∈ V)
7	6	expcom 413	. . . . . 6 ⊢ (𝐼 ∈ V → (𝐹 Fn 𝐼 → 𝐹 ∈ V))
8	5, 7	syl 17	. . . . 5 ⊢ (𝜑 → (𝐹 Fn 𝐼 → 𝐹 ∈ V))
9	8	adantrd 491	. . . 4 ⊢ (𝜑 → ((𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥)) → 𝐹 ∈ V))
10		fveq2 6907	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → (𝑆‘𝑖) = (𝑆‘𝑥))
11	10	cbvixpv 8954	. . . . . . . 8 ⊢ X𝑖 ∈ 𝐼 (𝑆‘𝑖) = X𝑥 ∈ 𝐼 (𝑆‘𝑥)
12	11	eleq2i 2831	. . . . . . 7 ⊢ (𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ↔ 𝐹 ∈ X𝑥 ∈ 𝐼 (𝑆‘𝑥))
13		elixp2 8940	. . . . . . 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 535	. . . . 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 379	. . 3 ⊢ (𝜑 → (𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥))))
19	18	anbi1d 631	. 2 ⊢ (𝜑 → ((𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∧ 𝐹 finSupp 0 ) ↔ ((𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥)) ∧ 𝐹 finSupp 0 )))
20		breq1 5151	. . 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 3698	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  {crab 3433  Vcvv 3478   class class class wbr 5148  dom cdm 5689   Fn wfn 6558  ‘cfv 6563  Xcixp 8936   finSupp cfsupp 9399   DProd cdprd 20028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-oprab 7435  df-mpo 7436  df-ixp 8937  df-dprd 20030

This theorem is referenced by:  dprdff  20047  dprdfcl  20048  dprdffsupp  20049  dprdsubg  20059