Theorem dprdw 20077

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 3476	. . . . 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 20068	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V)
6		fnex 7215	. . . . . . 7 ⊢ ((𝐹 Fn 𝐼 ∧ 𝐼 ∈ V) → 𝐹 ∈ V)
7	6	expcom 418	. . . . . 6 ⊢ (𝐼 ∈ V → (𝐹 Fn 𝐼 → 𝐹 ∈ V))
8	5, 7	syl 18	. . . . 5 ⊢ (𝜑 → (𝐹 Fn 𝐼 → 𝐹 ∈ V))
9	8	adantrd 496	. . . 4 ⊢ (𝜑 → ((𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥)) → 𝐹 ∈ V))
10		fveq2 6881	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → (𝑆‘𝑖) = (𝑆‘𝑥))
11	10	cbvixpv 8909	. . . . . . . 8 ⊢ X𝑖 ∈ 𝐼 (𝑆‘𝑖) = X𝑥 ∈ 𝐼 (𝑆‘𝑥)
12	11	eleq2i 2855	. . . . . . 7 ⊢ (𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ↔ 𝐹 ∈ X𝑥 ∈ 𝐼 (𝑆‘𝑥))
13		elixp2 8895	. . . . . . 7 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐼 (𝑆‘𝑥) ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥)))
14		3anass 1111	. . . . . . 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 300	. . . . . 6 ⊢ (𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ↔ (𝐹 ∈ V ∧ (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥))))
16	15	baib 544	. . . . 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 382	. . 3 ⊢ (𝜑 → (𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥))))
19	18	anbi1d 642	. 2 ⊢ (𝜑 → ((𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∧ 𝐹 finSupp 0 ) ↔ ((𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥)) ∧ 𝐹 finSupp 0 )))
20		breq1 5112	. . 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 3654	. 2 ⊢ (𝐹 ∈ 𝑊 ↔ (𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∧ 𝐹 finSupp 0 ))
23		df-3an 1105	. 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 317	1 ⊢ (𝜑 → (𝐹 ∈ 𝑊 ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥) ∧ 𝐹 finSupp 0 )))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  {crab 3416  Vcvv 3455   class class class wbr 5109  dom cdm 5661   Fn wfn 6531  ‘cfv 6536  Xcixp 8891   finSupp cfsupp 9317   DProd cdprd 20060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-oprab 7414  df-mpo 7415  df-ixp 8892  df-dprd 20062

This theorem is referenced by:  dprdff  20079  dprdfcl  20080  dprdffsupp  20081  dprdsubg  20091