Theorem elpt 23555

Description: Elementhood in the bases of a 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
elpt	⊢ (𝑆 ∈ 𝐵 ↔ ∃ℎ((ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (ℎ‘𝑦)))

Distinct variable groups:   𝑔,ℎ,𝑤,𝑥,𝑦,𝑧,𝐴   𝑔,𝐹,ℎ,𝑤,𝑥,𝑦,𝑧   𝑆,𝑔,ℎ,𝑥

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

Proof of Theorem elpt

Step	Hyp	Ref	Expression
1		ptbas.1	. . 3 ⊢ 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	eleq2i 2831	. 2 ⊢ (𝑆 ∈ 𝐵 ↔ 𝑆 ∈ {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))})
3		simpr 485	. . . . 5 ⊢ (((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) → 𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))
4		ixpexg 8860	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ V → X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ V)
5		fvexd 6842	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (𝑔‘𝑦) ∈ V)
6	4, 5	mprg 3059	. . . . 5 ⊢ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ V
7	3, 6	eqeltrdi 2847	. . . 4 ⊢ (((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) → 𝑆 ∈ V)
8	7	exlimiv 1937	. . 3 ⊢ (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) → 𝑆 ∈ V)
9		eqeq1 2743	. . . . 5 ⊢ (𝑥 = 𝑆 → (𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) ↔ 𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)))
10	9	anbi2d 636	. . . 4 ⊢ (𝑥 = 𝑆 → (((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) ↔ ((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))))
11	10	exbidv 1928	. . 3 ⊢ (𝑥 = 𝑆 → (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) ↔ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))))
12	8, 11	elab3 3624	. 2 ⊢ (𝑆 ∈ {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))} ↔ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)))
13		fneq1 6576	. . . . 5 ⊢ (𝑔 = ℎ → (𝑔 Fn 𝐴 ↔ ℎ Fn 𝐴))
14		fveq1 6826	. . . . . . 7 ⊢ (𝑔 = ℎ → (𝑔‘𝑦) = (ℎ‘𝑦))
15	14	eleq1d 2824	. . . . . 6 ⊢ (𝑔 = ℎ → ((𝑔‘𝑦) ∈ (𝐹‘𝑦) ↔ (ℎ‘𝑦) ∈ (𝐹‘𝑦)))
16	15	ralbidv 3162	. . . . 5 ⊢ (𝑔 = ℎ → (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ↔ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦)))
17	14	eqeq1d 2741	. . . . . . 7 ⊢ (𝑔 = ℎ → ((𝑔‘𝑦) = ∪ (𝐹‘𝑦) ↔ (ℎ‘𝑦) = ∪ (𝐹‘𝑦)))
18	17	rexralbidv 3205	. . . . . 6 ⊢ (𝑔 = ℎ → (∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦) ↔ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)))
19		difeq2 4051	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (𝐴 ∖ 𝑧) = (𝐴 ∖ 𝑤))
20	19	raleqdv 3297	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (∀𝑦 ∈ (𝐴 ∖ 𝑧)(ℎ‘𝑦) = ∪ (𝐹‘𝑦) ↔ ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)))
21	20	cbvrexvw 3218	. . . . . 6 ⊢ (∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(ℎ‘𝑦) = ∪ (𝐹‘𝑦) ↔ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))
22	18, 21	bitrdi 288	. . . . 5 ⊢ (𝑔 = ℎ → (∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦) ↔ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)))
23	13, 16, 22	3anbi123d 1444	. . . 4 ⊢ (𝑔 = ℎ → ((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ↔ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))))
24	14	ixpeq2dv 8851	. . . . 5 ⊢ (𝑔 = ℎ → X𝑦 ∈ 𝐴 (𝑔‘𝑦) = X𝑦 ∈ 𝐴 (ℎ‘𝑦))
25	24	eqeq2d 2750	. . . 4 ⊢ (𝑔 = ℎ → (𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) ↔ 𝑆 = X𝑦 ∈ 𝐴 (ℎ‘𝑦)))
26	23, 25	anbi12d 638	. . 3 ⊢ (𝑔 = ℎ → (((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) ↔ ((ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (ℎ‘𝑦))))
27	26	cbvexvw 2044	. 2 ⊢ (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) ↔ ∃ℎ((ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (ℎ‘𝑦)))
28	2, 12, 27	3bitri 298	1 ⊢ (𝑆 ∈ 𝐵 ↔ ∃ℎ((ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (ℎ‘𝑦)))

Syntax hints:   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2717  ∀wral 3053  ∃wrex 3063  Vcvv 3431   ∖ cdif 3880  ∪ cuni 4838   Fn wfn 6480  ‘cfv 6485  Xcixp 8835  Fincfn 8883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-ixp 8836

This theorem is referenced by:  elptr  23556  ptbasin  23560  ptbasfi  23564  ptrecube  37987