Theorem elptr2 23806

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 6893	. . . 4 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦)
2		nfcv 2924	. . . 4 ⊢ Ⅎ𝑦((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘)
3		fveq2 6882	. . . 4 ⊢ (𝑦 = 𝑘 → ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘))
4	1, 2, 3	cbvixp 8925	. . 3 ⊢ X𝑦 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = X𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘)
5		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
6		elptr2.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ∈ (𝐹‘𝑘))
7		eqid 2762	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝑆) = (𝑘 ∈ 𝐴 ↦ 𝑆)
8	7	fvmpt2 7002	. . . . 5 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝑆 ∈ (𝐹‘𝑘)) → ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = 𝑆)
9	5, 6, 8	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = 𝑆)
10	9	ixpeq2dva 8923	. . 3 ⊢ (𝜑 → X𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = X𝑘 ∈ 𝐴 𝑆)
11	4, 10	eqtrid 2809	. 2 ⊢ (𝜑 → X𝑦 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = X𝑘 ∈ 𝐴 𝑆)
12		elptr2.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
13	6	ralrimiva 3156	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝑆 ∈ (𝐹‘𝑘))
14	7	fnmpt 6676	. . . 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 2862	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) ∈ (𝐹‘𝑘))
17	16	ralrimiva 3156	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) ∈ (𝐹‘𝑘))
18	1	nfel1 2940	. . . . 5 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) ∈ (𝐹‘𝑦)
19		nfv 1947	. . . . 5 ⊢ Ⅎ𝑦((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) ∈ (𝐹‘𝑘)
20		fveq2 6882	. . . . . 6 ⊢ (𝑦 = 𝑘 → (𝐹‘𝑦) = (𝐹‘𝑘))
21	3, 20	eleq12d 2856	. . . . 5 ⊢ (𝑦 = 𝑘 → (((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) ∈ (𝐹‘𝑦) ↔ ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) ∈ (𝐹‘𝑘)))
22	18, 19, 21	cbvralw 3306	. . . 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 4081	. . . . . . 7 ⊢ (𝑘 ∈ (𝐴 ∖ 𝑊) → 𝑘 ∈ 𝐴)
26	25, 9	sylan2 605	. . . . . 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 2797	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = ∪ (𝐹‘𝑘))
29	28	ralrimiva 3156	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ (𝐴 ∖ 𝑊)((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = ∪ (𝐹‘𝑘))
30	1	nfeq1 2939	. . . . 5 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = ∪ (𝐹‘𝑦)
31		nfv 1947	. . . . 5 ⊢ Ⅎ𝑦((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = ∪ (𝐹‘𝑘)
32	20	unieqd 4883	. . . . . 6 ⊢ (𝑦 = 𝑘 → ∪ (𝐹‘𝑦) = ∪ (𝐹‘𝑘))
33	3, 32	eqeq12d 2778	. . . . 5 ⊢ (𝑦 = 𝑘 → (((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = ∪ (𝐹‘𝑦) ↔ ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = ∪ (𝐹‘𝑘)))
34	30, 31, 33	cbvralw 3306	. . . 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 23805	. . 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 1406	. 2 ⊢ (𝜑 → X𝑦 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) ∈ 𝐵)
39	11, 38	eqeltrrd 2863	1 ⊢ (𝜑 → X𝑘 ∈ 𝐴 𝑆 ∈ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2740  ∀wral 3078  ∃wrex 3088   ∖ cdif 3899  ∪ cuni 4870   ↦ cmpt 5190   Fn wfn 6532  ‘cfv 6537  Xcixp 8908  Fincfn 8956

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ixp 8909

This theorem is used by:  ptbasid  23807  ptbasin  23809  ptpjpre2  23812  ptopn  23815