Theorem dprdw 18617

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 3364	. . . . 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 18608	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V)
6		fnex 6625	. . . . . . 7 ⊢ ((𝐹 Fn 𝐼 ∧ 𝐼 ∈ V) → 𝐹 ∈ V)
7	6	expcom 398	. . . . . 6 ⊢ (𝐼 ∈ V → (𝐹 Fn 𝐼 → 𝐹 ∈ V))
8	5, 7	syl 17	. . . . 5 ⊢ (𝜑 → (𝐹 Fn 𝐼 → 𝐹 ∈ V))
9	8	adantrd 479	. . . 4 ⊢ (𝜑 → ((𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥)) → 𝐹 ∈ V))
10		fveq2 6332	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → (𝑆‘𝑖) = (𝑆‘𝑥))
11	10	cbvixpv 8080	. . . . . . . 8 ⊢ X𝑖 ∈ 𝐼 (𝑆‘𝑖) = X𝑥 ∈ 𝐼 (𝑆‘𝑥)
12	11	eleq2i 2842	. . . . . . 7 ⊢ (𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ↔ 𝐹 ∈ X𝑥 ∈ 𝐼 (𝑆‘𝑥))
13		elixp2 8066	. . . . . . 7 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐼 (𝑆‘𝑥) ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥)))
14		3anass 1080	. . . . . . 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 286	. . . . . 6 ⊢ (𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ↔ (𝐹 ∈ V ∧ (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥))))
16	15	baib 525	. . . . 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 368	. . 3 ⊢ (𝜑 → (𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥))))
19	18	anbi1d 615	. 2 ⊢ (𝜑 → ((𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∧ 𝐹 finSupp 0 ) ↔ ((𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥)) ∧ 𝐹 finSupp 0 )))
20		breq1 4789	. . 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 3518	. 2 ⊢ (𝐹 ∈ 𝑊 ↔ (𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∧ 𝐹 finSupp 0 ))
23		df-3an 1073	. 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 303	1 ⊢ (𝜑 → (𝐹 ∈ 𝑊 ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥) ∧ 𝐹 finSupp 0 )))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ∀wral 3061  {crab 3065  Vcvv 3351   class class class wbr 4786  dom cdm 5249   Fn wfn 6026  ‘cfv 6031  Xcixp 8062   finSupp cfsupp 8431   DProd cdprd 18600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-oprab 6797  df-mpt2 6798  df-ixp 8063  df-dprd 18602

This theorem is referenced by:  dprdff  18619  dprdfcl  18620  dprdffsupp  18621  dprdsubg  18631