Theorem ptbasin2 23494

Description: The basis for a product topology is closed under intersections. (Contributed by Mario Carneiro, 19-Mar-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
ptbasin2	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (fi‘𝐵) = 𝐵)

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

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

Proof of Theorem ptbasin2

Dummy variables 𝑘 𝑢 𝑣 are mutually distinct and distinct from all other variables.

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	ptbasin 23493	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢 ∩ 𝑣) ∈ 𝐵)
3	2	ralrimivva 3175	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ∩ 𝑣) ∈ 𝐵)
4	1	ptuni2 23492	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐵)
5		ixpexg 8846	. . . . . 6 ⊢ (∀𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ V → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ V)
6		fvex 6835	. . . . . . . 8 ⊢ (𝐹‘𝑘) ∈ V
7	6	uniex 7674	. . . . . . 7 ⊢ ∪ (𝐹‘𝑘) ∈ V
8	7	a1i 11	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 → ∪ (𝐹‘𝑘) ∈ V)
9	5, 8	mprg 3053	. . . . 5 ⊢ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ V
10	4, 9	eqeltrrdi 2840	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∪ 𝐵 ∈ V)
11		uniexb 7697	. . . 4 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V)
12	10, 11	sylibr 234	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 ∈ V)
13		inficl 9309	. . 3 ⊢ (𝐵 ∈ V → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ∩ 𝑣) ∈ 𝐵 ↔ (fi‘𝐵) = 𝐵))
14	12, 13	syl 17	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ∩ 𝑣) ∈ 𝐵 ↔ (fi‘𝐵) = 𝐵))
15	3, 14	mpbid 232	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (fi‘𝐵) = 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ∖ cdif 3899   ∩ cin 3901  ∪ cuni 4859   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  Xcixp 8821  Fincfn 8869  ficfi 9294  Topctop 22809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-om 7797  df-1o 8385  df-2o 8386  df-ixp 8822  df-en 8870  df-fin 8873  df-fi 9295  df-top 22810

This theorem is referenced by:  ptbas  23495  ptbasfi  23497