Theorem elptr2 22725

Description: A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)

Hypotheses

Ref	Expression
ptbas.1	⊢ 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}
elptr2.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
elptr2.2	⊢ (𝜑 → 𝑊 ∈ Fin)
elptr2.3	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ∈ (𝐹‘𝑘))
elptr2.4	⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝑆 = ∪ (𝐹‘𝑘))

Assertion

Ref	Expression
elptr2	⊢ (𝜑 → X𝑘 ∈ 𝐴 𝑆 ∈ 𝐵)

Distinct variable groups:   𝐵,𝑘   𝑥,𝑔,𝑦   𝜑,𝑘   𝑔,𝑘,𝑧,𝐴,𝑥,𝑦   𝑔,𝐹,𝑘,𝑥,𝑦,𝑧   𝑆,𝑔,𝑥   𝑔,𝑉,𝑘,𝑥,𝑦,𝑧   𝑘,𝑊,𝑦   𝑦,𝑆

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑔)   𝐵(𝑥,𝑦,𝑧,𝑔)   𝑆(𝑧,𝑘)   𝑊(𝑥,𝑧,𝑔)

Proof of Theorem elptr2

Step	Hyp	Ref	Expression
1		nffvmpt1 6785	. . . 4 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦)
2		nfcv 2907	. . . 4 ⊢ Ⅎ𝑦((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘)
3		fveq2 6774	. . . 4 ⊢ (𝑦 = 𝑘 → ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘))
4	1, 2, 3	cbvixp 8702	. . 3 ⊢ X𝑦 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = X𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘)
5		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
6		elptr2.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ∈ (𝐹‘𝑘))
7		eqid 2738	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝑆) = (𝑘 ∈ 𝐴 ↦ 𝑆)
8	7	fvmpt2 6886	. . . . 5 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝑆 ∈ (𝐹‘𝑘)) → ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = 𝑆)
9	5, 6, 8	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = 𝑆)
10	9	ixpeq2dva 8700	. . 3 ⊢ (𝜑 → X𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = X𝑘 ∈ 𝐴 𝑆)
11	4, 10	eqtrid 2790	. 2 ⊢ (𝜑 → X𝑦 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = X𝑘 ∈ 𝐴 𝑆)
12		elptr2.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
13	6	ralrimiva 3103	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝑆 ∈ (𝐹‘𝑘))
14	7	fnmpt 6573	. . . 4 ⊢ (∀𝑘 ∈ 𝐴 𝑆 ∈ (𝐹‘𝑘) → (𝑘 ∈ 𝐴 ↦ 𝑆) Fn 𝐴)
15	13, 14	syl 17	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑆) Fn 𝐴)
16	9, 6	eqeltrd 2839	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) ∈ (𝐹‘𝑘))
17	16	ralrimiva 3103	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) ∈ (𝐹‘𝑘))
18	1	nfel1 2923	. . . . 5 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) ∈ (𝐹‘𝑦)
19		nfv 1917	. . . . 5 ⊢ Ⅎ𝑦((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) ∈ (𝐹‘𝑘)
20		fveq2 6774	. . . . . 6 ⊢ (𝑦 = 𝑘 → (𝐹‘𝑦) = (𝐹‘𝑘))
21	3, 20	eleq12d 2833	. . . . 5 ⊢ (𝑦 = 𝑘 → (((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) ∈ (𝐹‘𝑦) ↔ ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) ∈ (𝐹‘𝑘)))
22	18, 19, 21	cbvralw 3373	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) ∈ (𝐹‘𝑦) ↔ ∀𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) ∈ (𝐹‘𝑘))
23	17, 22	sylibr 233	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) ∈ (𝐹‘𝑦))
24		elptr2.2	. . 3 ⊢ (𝜑 → 𝑊 ∈ Fin)
25		eldifi 4061	. . . . . . 7 ⊢ (𝑘 ∈ (𝐴 ∖ 𝑊) → 𝑘 ∈ 𝐴)
26	25, 9	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = 𝑆)
27		elptr2.4	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝑆 = ∪ (𝐹‘𝑘))
28	26, 27	eqtrd 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = ∪ (𝐹‘𝑘))
29	28	ralrimiva 3103	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ (𝐴 ∖ 𝑊)((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = ∪ (𝐹‘𝑘))
30	1	nfeq1 2922	. . . . 5 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = ∪ (𝐹‘𝑦)
31		nfv 1917	. . . . 5 ⊢ Ⅎ𝑦((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = ∪ (𝐹‘𝑘)
32	20	unieqd 4853	. . . . . 6 ⊢ (𝑦 = 𝑘 → ∪ (𝐹‘𝑦) = ∪ (𝐹‘𝑘))
33	3, 32	eqeq12d 2754	. . . . 5 ⊢ (𝑦 = 𝑘 → (((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = ∪ (𝐹‘𝑦) ↔ ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = ∪ (𝐹‘𝑘)))
34	30, 31, 33	cbvralw 3373	. . . 4 ⊢ (∀𝑦 ∈ (𝐴 ∖ 𝑊)((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = ∪ (𝐹‘𝑦) ↔ ∀𝑘 ∈ (𝐴 ∖ 𝑊)((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = ∪ (𝐹‘𝑘))
35	29, 34	sylibr 233	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴 ∖ 𝑊)((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = ∪ (𝐹‘𝑦))
36		ptbas.1	. . . 4 ⊢ 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}
37	36	elptr 22724	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝑘 ∈ 𝐴 ↦ 𝑆) Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) ∈ (𝐹‘𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑊)((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = ∪ (𝐹‘𝑦))) → X𝑦 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) ∈ 𝐵)
38	12, 15, 23, 24, 35, 37	syl122anc 1378	. 2 ⊢ (𝜑 → X𝑦 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) ∈ 𝐵)
39	11, 38	eqeltrrd 2840	1 ⊢ (𝜑 → X𝑘 ∈ 𝐴 𝑆 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715  ∀wral 3064  ∃wrex 3065   ∖ cdif 3884  ∪ cuni 4839   ↦ cmpt 5157   Fn wfn 6428  ‘cfv 6433  Xcixp 8685  Fincfn 8733

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-pow 5288  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-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-pw 4535  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-ixp 8686

This theorem is referenced by:  ptbasid  22726  ptbasin  22728  ptpjpre2  22731  ptopn  22734