Theorem ptbasin2 23607

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 23606	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢 ∩ 𝑣) ∈ 𝐵)
3	2	ralrimivva 3208	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ∩ 𝑣) ∈ 𝐵)
4	1	ptuni2 23605	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐵)
5		ixpexg 8980	. . . . . 6 ⊢ (∀𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ V → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ V)
6		fvex 6933	. . . . . . . 8 ⊢ (𝐹‘𝑘) ∈ V
7	6	uniex 7776	. . . . . . 7 ⊢ ∪ (𝐹‘𝑘) ∈ V
8	7	a1i 11	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 → ∪ (𝐹‘𝑘) ∈ V)
9	5, 8	mprg 3073	. . . . 5 ⊢ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ V
10	4, 9	eqeltrrdi 2853	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∪ 𝐵 ∈ V)
11		uniexb 7799	. . . 4 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V)
12	10, 11	sylibr 234	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 ∈ V)
13		inficl 9494	. . 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 1087   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ∖ cdif 3973   ∩ cin 3975  ∪ cuni 4931   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  Xcixp 8955  Fincfn 9003  ficfi 9479  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-int 4971  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-suc 6401  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-1o 8522  df-2o 8523  df-ixp 8956  df-en 9004  df-fin 9007  df-fi 9480  df-top 22921

This theorem is referenced by:  ptbas  23608  ptbasfi  23610