Theorem ptuni2 23950

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 23949	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ 𝐵)
3		elssuni 4904	. . 3 ⊢ (X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ 𝐵 → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ⊆ ∪ 𝐵)
4	2, 3	syl 18	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ⊆ ∪ 𝐵)
5		simpr2 1214	. . . . . . . . . 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 4904	. . . . . . . . . . 11 ⊢ ((𝑔‘𝑦) ∈ (𝐹‘𝑦) → (𝑔‘𝑦) ⊆ ∪ (𝐹‘𝑦))
7	6	ralimi 3103	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) → ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ ∪ (𝐹‘𝑦))
8		ss2ixp 8976	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ ∪ (𝐹‘𝑦) → X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦))
9	5, 7, 8	3syl 19	. . . . . . . . 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 6916	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑘 → (𝐹‘𝑦) = (𝐹‘𝑘))
11	10	unieqd 4885	. . . . . . . . . 10 ⊢ (𝑦 = 𝑘 → ∪ (𝐹‘𝑦) = ∪ (𝐹‘𝑘))
12	11	cbvixpv 8981	. . . . . . . . 9 ⊢ X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦) = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)
13	9, 12	sseqtrdi 3953	. . . . . . . 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 4565	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ 𝑥 ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
15		sseq1 3938	. . . . . . . . 9 ⊢ (𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (𝑥 ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)))
16	14, 15	bitrid 286	. . . . . . . 8 ⊢ (𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (𝑥 ∈ 𝒫 X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)))
17	13, 16	syl5ibrcom 250	. . . . . . 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 459	. . . . . 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 1968	. . . . 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 3995	. . . 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 3951	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 ⊆ 𝒫 X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
22		sspwuni 5065	. . 3 ⊢ (𝐵 ⊆ 𝒫 X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ ∪ 𝐵 ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
23	21, 22	sylib 221	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∪ 𝐵 ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
24	4, 23	eqssd 3930	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∖ cdif 4048  𝒫 cpw 4560  ∪ cuni 4872   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  Xcixp 8963  Fincfn 9011  Topctop 23266

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-lim 6393  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-om 7912  df-ixp 8964  df-en 9012  df-fin 9015  df-top 23267

This theorem is used by:  ptbasin2  23952  ptbasfi  23955  ptuni  23968