Theorem elptr2 22184

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 6683	. . . 4 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦)
2		nfcv 2979	. . . 4 ⊢ Ⅎ𝑦((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘)
3		fveq2 6672	. . . 4 ⊢ (𝑦 = 𝑘 → ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘))
4	1, 2, 3	cbvixp 8480	. . 3 ⊢ X𝑦 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = X𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘)
5		simpr 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
6		elptr2.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ∈ (𝐹‘𝑘))
7		eqid 2823	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝑆) = (𝑘 ∈ 𝐴 ↦ 𝑆)
8	7	fvmpt2 6781	. . . . 5 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝑆 ∈ (𝐹‘𝑘)) → ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = 𝑆)
9	5, 6, 8	syl2anc 586	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = 𝑆)
10	9	ixpeq2dva 8478	. . 3 ⊢ (𝜑 → X𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = X𝑘 ∈ 𝐴 𝑆)
11	4, 10	syl5eq 2870	. 2 ⊢ (𝜑 → X𝑦 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = X𝑘 ∈ 𝐴 𝑆)
12		elptr2.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
13	6	ralrimiva 3184	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝑆 ∈ (𝐹‘𝑘))
14	7	fnmpt 6490	. . . 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 2915	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) ∈ (𝐹‘𝑘))
17	16	ralrimiva 3184	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) ∈ (𝐹‘𝑘))
18	1	nfel1 2996	. . . . 5 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) ∈ (𝐹‘𝑦)
19		nfv 1915	. . . . 5 ⊢ Ⅎ𝑦((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) ∈ (𝐹‘𝑘)
20		fveq2 6672	. . . . . 6 ⊢ (𝑦 = 𝑘 → (𝐹‘𝑦) = (𝐹‘𝑘))
21	3, 20	eleq12d 2909	. . . . 5 ⊢ (𝑦 = 𝑘 → (((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) ∈ (𝐹‘𝑦) ↔ ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) ∈ (𝐹‘𝑘)))
22	18, 19, 21	cbvralw 3443	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) ∈ (𝐹‘𝑦) ↔ ∀𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) ∈ (𝐹‘𝑘))
23	17, 22	sylibr 236	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) ∈ (𝐹‘𝑦))
24		elptr2.2	. . 3 ⊢ (𝜑 → 𝑊 ∈ Fin)
25		eldifi 4105	. . . . . . 7 ⊢ (𝑘 ∈ (𝐴 ∖ 𝑊) → 𝑘 ∈ 𝐴)
26	25, 9	sylan2 594	. . . . . 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 2858	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = ∪ (𝐹‘𝑘))
29	28	ralrimiva 3184	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ (𝐴 ∖ 𝑊)((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = ∪ (𝐹‘𝑘))
30	1	nfeq1 2995	. . . . 5 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = ∪ (𝐹‘𝑦)
31		nfv 1915	. . . . 5 ⊢ Ⅎ𝑦((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = ∪ (𝐹‘𝑘)
32	20	unieqd 4854	. . . . . 6 ⊢ (𝑦 = 𝑘 → ∪ (𝐹‘𝑦) = ∪ (𝐹‘𝑘))
33	3, 32	eqeq12d 2839	. . . . 5 ⊢ (𝑦 = 𝑘 → (((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = ∪ (𝐹‘𝑦) ↔ ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = ∪ (𝐹‘𝑘)))
34	30, 31, 33	cbvralw 3443	. . . 4 ⊢ (∀𝑦 ∈ (𝐴 ∖ 𝑊)((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) = ∪ (𝐹‘𝑦) ↔ ∀𝑘 ∈ (𝐴 ∖ 𝑊)((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑘) = ∪ (𝐹‘𝑘))
35	29, 34	sylibr 236	. . 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 22183	. . 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 1375	. 2 ⊢ (𝜑 → X𝑦 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝑆)‘𝑦) ∈ 𝐵)
39	11, 38	eqeltrrd 2916	1 ⊢ (𝜑 → X𝑘 ∈ 𝐴 𝑆 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537  ∃wex 1780   ∈ wcel 2114  {cab 2801  ∀wral 3140  ∃wrex 3141   ∖ cdif 3935  ∪ cuni 4840   ↦ cmpt 5148   Fn wfn 6352  ‘cfv 6357  Xcixp 8463  Fincfn 8511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ixp 8464

This theorem is referenced by:  ptbasid  22185  ptbasin  22187  ptpjpre2  22190  ptopn  22193