Theorem ptbasin2 23716

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 23715	. . 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 23714	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐵)
5		ixpexg 8921	. . . . . 6 ⊢ (∀𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ V → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ V)
6		fvex 6896	. . . . . . . 8 ⊢ (𝐹‘𝑘) ∈ V
7	6	uniex 7741	. . . . . . 7 ⊢ ∪ (𝐹‘𝑘) ∈ V
8	7	a1i 11	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 → ∪ (𝐹‘𝑘) ∈ V)
9	5, 8	mprg 3085	. . . . 5 ⊢ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ V
10	4, 9	eqeltrrdi 2872	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∪ 𝐵 ∈ V)
11		uniexb 7764	. . . 4 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V)
12	10, 11	sylibr 237	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 ∈ V)
13		inficl 9386	. . 3 ⊢ (𝐵 ∈ V → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ∩ 𝑣) ∈ 𝐵 ↔ (fi‘𝐵) = 𝐵))
14	12, 13	syl 18	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ∩ 𝑣) ∈ 𝐵 ↔ (fi‘𝐵) = 𝐵))
15	3, 14	mpbid 235	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (fi‘𝐵) = 𝐵)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ∖ cdif 3903   ∩ cin 3905  ∪ cuni 4873   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  Xcixp 8896  Fincfn 8944  ficfi 9371  Topctop 23031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-om 7864  df-1o 8454  df-2o 8455  df-ixp 8897  df-en 8945  df-fin 8948  df-fi 9372  df-top 23032

This theorem is referenced by:  ptbas  23717  ptbasfi  23719