Theorem ptuni2 23532

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 23531	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ 𝐵)
3		elssuni 4896	. . 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 1197	. . . . . . . . . 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 4896	. . . . . . . . . . 11 ⊢ ((𝑔‘𝑦) ∈ (𝐹‘𝑦) → (𝑔‘𝑦) ⊆ ∪ (𝐹‘𝑦))
7	6	ralimi 3075	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) → ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ ∪ (𝐹‘𝑦))
8		ss2ixp 8860	. . . . . . . . . 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 6842	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑘 → (𝐹‘𝑦) = (𝐹‘𝑘))
11	10	unieqd 4878	. . . . . . . . . 10 ⊢ (𝑦 = 𝑘 → ∪ (𝐹‘𝑦) = ∪ (𝐹‘𝑘))
12	11	cbvixpv 8865	. . . . . . . . 9 ⊢ X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦) = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)
13	9, 12	sseqtrdi 3976	. . . . . . . 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 4561	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ 𝑥 ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
15		sseq1 3961	. . . . . . . . 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 1935	. . . . 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 4021	. . . 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 3974	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 ⊆ 𝒫 X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
22		sspwuni 5057	. . 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 3953	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062   ∖ cdif 3900   ⊆ wss 3903  𝒫 cpw 4556  ∪ cuni 4865   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  Xcixp 8847  Fincfn 8895  Topctop 22849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-om 7819  df-ixp 8848  df-en 8896  df-fin 8899  df-top 22850

This theorem is referenced by:  ptbasin2  23534  ptbasfi  23537  ptuni  23550