Theorem ptuni2 23463

Description: The base set for the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)

Hypothesis

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))}

Assertion

Ref	Expression
ptuni2	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐵)

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

Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧,𝑔)

Proof of Theorem ptuni2

Step	Hyp	Ref	Expression
1		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))}
2	1	ptbasid 23462	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ 𝐵)
3		elssuni 4901	. . 3 ⊢ (X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ 𝐵 → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ⊆ ∪ 𝐵)
4	2, 3	syl 17	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ⊆ ∪ 𝐵)
5		simpr2 1196	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) → ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦))
6		elssuni 4901	. . . . . . . . . . 11 ⊢ ((𝑔‘𝑦) ∈ (𝐹‘𝑦) → (𝑔‘𝑦) ⊆ ∪ (𝐹‘𝑦))
7	6	ralimi 3066	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) → ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ ∪ (𝐹‘𝑦))
8		ss2ixp 8883	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ ∪ (𝐹‘𝑦) → X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦))
9	5, 7, 8	3syl 18	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) → X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦))
10		fveq2 6858	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑘 → (𝐹‘𝑦) = (𝐹‘𝑘))
11	10	unieqd 4884	. . . . . . . . . 10 ⊢ (𝑦 = 𝑘 → ∪ (𝐹‘𝑦) = ∪ (𝐹‘𝑘))
12	11	cbvixpv 8888	. . . . . . . . 9 ⊢ X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦) = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)
13	9, 12	sseqtrdi 3987	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) → X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
14		velpw 4568	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ 𝑥 ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
15		sseq1 3972	. . . . . . . . 9 ⊢ (𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (𝑥 ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)))
16	14, 15	bitrid 283	. . . . . . . 8 ⊢ (𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (𝑥 ∈ 𝒫 X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)))
17	13, 16	syl5ibrcom 247	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) → (𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → 𝑥 ∈ 𝒫 X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)))
18	17	expimpd 453	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) → 𝑥 ∈ 𝒫 X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)))
19	18	exlimdv 1933	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) → 𝑥 ∈ 𝒫 X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)))
20	19	abssdv 4031	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → {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))} ⊆ 𝒫 X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
21	1, 20	eqsstrid 3985	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 ⊆ 𝒫 X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
22		sspwuni 5064	. . 3 ⊢ (𝐵 ⊆ 𝒫 X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ ∪ 𝐵 ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
23	21, 22	sylib 218	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∪ 𝐵 ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
24	4, 23	eqssd 3964	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053   ∖ cdif 3911   ⊆ wss 3914  𝒫 cpw 4563  ∪ cuni 4871   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  Xcixp 8870  Fincfn 8918  Topctop 22780

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-om 7843  df-ixp 8871  df-en 8919  df-fin 8922  df-top 22781

This theorem is referenced by:  ptbasin2  23465  ptbasfi  23468  ptuni  23481