Theorem ptuni2 23531

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 23530	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ 𝐵)
3		elssuni 4917	. . 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 1195	. . . . . . . . . 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 4917	. . . . . . . . . . 11 ⊢ ((𝑔‘𝑦) ∈ (𝐹‘𝑦) → (𝑔‘𝑦) ⊆ ∪ (𝐹‘𝑦))
7	6	ralimi 3072	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) → ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ ∪ (𝐹‘𝑦))
8		ss2ixp 8932	. . . . . . . . . 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 6886	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑘 → (𝐹‘𝑦) = (𝐹‘𝑘))
11	10	unieqd 4900	. . . . . . . . . 10 ⊢ (𝑦 = 𝑘 → ∪ (𝐹‘𝑦) = ∪ (𝐹‘𝑘))
12	11	cbvixpv 8937	. . . . . . . . 9 ⊢ X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦) = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)
13	9, 12	sseqtrdi 4004	. . . . . . . 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 4585	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ 𝑥 ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
15		sseq1 3989	. . . . . . . . 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 1932	. . . . 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 4048	. . . 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 4002	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 ⊆ 𝒫 X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
22		sspwuni 5080	. . 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 3981	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539  ∃wex 1778   ∈ wcel 2107  {cab 2712  ∀wral 3050  ∃wrex 3059   ∖ cdif 3928   ⊆ wss 3931  𝒫 cpw 4580  ∪ cuni 4887   Fn wfn 6536  ⟶wf 6537  ‘cfv 6541  Xcixp 8919  Fincfn 8967  Topctop 22848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-ord 6366  df-on 6367  df-lim 6368  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-om 7870  df-ixp 8920  df-en 8968  df-fin 8971  df-top 22849

This theorem is referenced by:  ptbasin2  23533  ptbasfi  23536  ptuni  23549