Theorem ptuni2 23605

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 23604	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ 𝐵)
3		elssuni 4961	. . 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 4961	. . . . . . . . . . 11 ⊢ ((𝑔‘𝑦) ∈ (𝐹‘𝑦) → (𝑔‘𝑦) ⊆ ∪ (𝐹‘𝑦))
7	6	ralimi 3089	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) → ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ ∪ (𝐹‘𝑦))
8		ss2ixp 8968	. . . . . . . . . 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 6920	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑘 → (𝐹‘𝑦) = (𝐹‘𝑘))
11	10	unieqd 4944	. . . . . . . . . 10 ⊢ (𝑦 = 𝑘 → ∪ (𝐹‘𝑦) = ∪ (𝐹‘𝑘))
12	11	cbvixpv 8973	. . . . . . . . 9 ⊢ X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦) = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)
13	9, 12	sseqtrdi 4059	. . . . . . . 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 4627	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ 𝑥 ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
15		sseq1 4034	. . . . . . . . 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 4091	. . . 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 4057	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 ⊆ 𝒫 X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
22		sspwuni 5123	. . 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 4026	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076   ∖ cdif 3973   ⊆ wss 3976  𝒫 cpw 4622  ∪ cuni 4931   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  Xcixp 8955  Fincfn 9003  Topctop 22920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-om 7904  df-ixp 8956  df-en 9004  df-fin 9007  df-top 22921

This theorem is referenced by:  ptbasin2  23607  ptbasfi  23610  ptuni  23623