Theorem dprdw 19918

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 3465	. . . . 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 19909	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V)
6		fnex 7173	. . . . . . 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 6840	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → (𝑆‘𝑖) = (𝑆‘𝑥))
11	10	cbvixpv 8865	. . . . . . . 8 ⊢ X𝑖 ∈ 𝐼 (𝑆‘𝑖) = X𝑥 ∈ 𝐼 (𝑆‘𝑥)
12	11	eleq2i 2820	. . . . . . 7 ⊢ (𝐹 ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ↔ 𝐹 ∈ X𝑥 ∈ 𝐼 (𝑆‘𝑥))
13		elixp2 8851	. . . . . . 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 5105	. . 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 3659	. 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 1540   ∈ wcel 2109  ∀wral 3044  {crab 3402  Vcvv 3444   class class class wbr 5102  dom cdm 5631   Fn wfn 6494  ‘cfv 6499  Xcixp 8847   finSupp cfsupp 9288   DProd cdprd 19901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-oprab 7373  df-mpo 7374  df-ixp 8848  df-dprd 19903

This theorem is referenced by:  dprdff  19920  dprdfcl  19921  dprdffsupp  19922  dprdsubg  19932