Theorem elptr2 23692

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 6882	. . . 4 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦)
2		nfcv 2927	. . . 4 ⊢ Ⅎ𝑦((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘)
3		fveq2 6871	. . . 4 ⊢ (𝑦 = 𝑘 → ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘))
4	1, 2, 3	cbvixp 8900	. . 3 ⊢ X𝑦 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = X𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘)
5		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
6		elptr2.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ∈ (𝐹‘𝑘))
7		eqid 2765	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝑆) = (𝑘 ∈ 𝐴 ↦ 𝑆)
8	7	fvmpt2 6991	. . . . 5 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝑆 ∈ (𝐹‘𝑘)) → ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = 𝑆)
9	5, 6, 8	syl2anc 595	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = 𝑆)
10	9	ixpeq2dva 8898	. . 3 ⊢ (𝜑 → X𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = X𝑘 ∈ 𝐴 𝑆)
11	4, 10	eqtrid 2812	. 2 ⊢ (𝜑 → X𝑦 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = X𝑘 ∈ 𝐴 𝑆)
12		elptr2.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
13	6	ralrimiva 3157	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝑆 ∈ (𝐹‘𝑘))
14	7	fnmpt 6665	. . . 4 ⊢ (∀𝑘 ∈ 𝐴 𝑆 ∈ (𝐹‘𝑘) → (𝑘 ∈ 𝐴 ↦ 𝑆) Fn 𝐴)
15	13, 14	syl 18	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑆) Fn 𝐴)
16	9, 6	eqeltrd 2865	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) ∈ (𝐹‘𝑘))
17	16	ralrimiva 3157	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) ∈ (𝐹‘𝑘))
18	1	nfel1 2943	. . . . 5 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) ∈ (𝐹‘𝑦)
19		nfv 1937	. . . . 5 ⊢ Ⅎ𝑦((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) ∈ (𝐹‘𝑘)
20		fveq2 6871	. . . . . 6 ⊢ (𝑦 = 𝑘 → (𝐹‘𝑦) = (𝐹‘𝑘))
21	3, 20	eleq12d 2859	. . . . 5 ⊢ (𝑦 = 𝑘 → (((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) ∈ (𝐹‘𝑦) ↔ ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) ∈ (𝐹‘𝑘)))
22	18, 19, 21	cbvralw 3307	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) ∈ (𝐹‘𝑦) ↔ ∀𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) ∈ (𝐹‘𝑘))
23	17, 22	sylibr 237	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) ∈ (𝐹‘𝑦))
24		elptr2.2	. . 3 ⊢ (𝜑 → 𝑊 ∈ Fin)
25		eldifi 4087	. . . . . . 7 ⊢ (𝑘 ∈ (𝐴 ∖ 𝑊) → 𝑘 ∈ 𝐴)
26	25, 9	sylan2 604	. . . . . 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 2800	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = ∪ (𝐹‘𝑘))
29	28	ralrimiva 3157	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ (𝐴 ∖ 𝑊)((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = ∪ (𝐹‘𝑘))
30	1	nfeq1 2942	. . . . 5 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = ∪ (𝐹‘𝑦)
31		nfv 1937	. . . . 5 ⊢ Ⅎ𝑦((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = ∪ (𝐹‘𝑘)
32	20	unieqd 4881	. . . . . 6 ⊢ (𝑦 = 𝑘 → ∪ (𝐹‘𝑦) = ∪ (𝐹‘𝑘))
33	3, 32	eqeq12d 2781	. . . . 5 ⊢ (𝑦 = 𝑘 → (((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = ∪ (𝐹‘𝑦) ↔ ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = ∪ (𝐹‘𝑘)))
34	30, 31, 33	cbvralw 3307	. . . 4 ⊢ (∀𝑦 ∈ (𝐴 ∖ 𝑊)((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = ∪ (𝐹‘𝑦) ↔ ∀𝑘 ∈ (𝐴 ∖ 𝑊)((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = ∪ (𝐹‘𝑘))
35	29, 34	sylibr 237	. . 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 23691	. . 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 1402	. 2 ⊢ (𝜑 → X𝑦 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) ∈ 𝐵)
39	11, 38	eqeltrrd 2866	1 ⊢ (𝜑 → X𝑘 ∈ 𝐴 𝑆 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563  ∃wex 1802   ∈ wcel 2145  {cab 2743  ∀wral 3079  ∃wrex 3089   ∖ cdif 3904  ∪ cuni 4868   ↦ cmpt 5186   Fn wfn 6520  ‘cfv 6525  Xcixp 8883  Fincfn 8931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ixp 8884

This theorem is referenced by:  ptbasid  23693  ptbasin  23695  ptpjpre2  23698  ptopn  23701