Theorem elpt 23694

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 2861	. 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 489	. . . . 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 8916	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ V → X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ V)
5		fvexd 6894	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (𝑔‘𝑦) ∈ V)
6	4, 5	mprg 3091	. . . . 5 ⊢ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ V
7	3, 6	eqeltrdi 2877	. . . 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 1957	. . 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 2773	. . . . 5 ⊢ (𝑥 = 𝑆 → (𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) ↔ 𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)))
10	9	anbi2d 641	. . . 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 1948	. . 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 3654	. 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 6624	. . . . 5 ⊢ (𝑔 = ℎ → (𝑔 Fn 𝐴 ↔ ℎ Fn 𝐴))
14		fveq1 6878	. . . . . . 7 ⊢ (𝑔 = ℎ → (𝑔‘𝑦) = (ℎ‘𝑦))
15	14	eleq1d 2854	. . . . . 6 ⊢ (𝑔 = ℎ → ((𝑔‘𝑦) ∈ (𝐹‘𝑦) ↔ (ℎ‘𝑦) ∈ (𝐹‘𝑦)))
16	15	ralbidv 3194	. . . . 5 ⊢ (𝑔 = ℎ → (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ↔ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦)))
17	14	eqeq1d 2771	. . . . . . 7 ⊢ (𝑔 = ℎ → ((𝑔‘𝑦) = ∪ (𝐹‘𝑦) ↔ (ℎ‘𝑦) = ∪ (𝐹‘𝑦)))
18	17	rexralbidv 3237	. . . . . 6 ⊢ (𝑔 = ℎ → (∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦) ↔ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)))
19		difeq2 4083	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (𝐴 ∖ 𝑧) = (𝐴 ∖ 𝑤))
20	19	raleqdv 3329	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (∀𝑦 ∈ (𝐴 ∖ 𝑧)(ℎ‘𝑦) = ∪ (𝐹‘𝑦) ↔ ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)))
21	20	cbvrexvw 3250	. . . . . 6 ⊢ (∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(ℎ‘𝑦) = ∪ (𝐹‘𝑦) ↔ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))
22	18, 21	bitrdi 290	. . . . 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 1462	. . . 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 8907	. . . . 5 ⊢ (𝑔 = ℎ → X𝑦 ∈ 𝐴 (𝑔‘𝑦) = X𝑦 ∈ 𝐴 (ℎ‘𝑦))
25	24	eqeq2d 2780	. . . 4 ⊢ (𝑔 = ℎ → (𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) ↔ 𝑆 = X𝑦 ∈ 𝐴 (ℎ‘𝑦)))
26	23, 25	anbi12d 643	. . 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 2064	. 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 300	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 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747  ∀wral 3085  ∃wrex 3095  Vcvv 3463   ∖ cdif 3910  ∪ cuni 4873   Fn wfn 6528  ‘cfv 6533  Xcixp 8891  Fincfn 8939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ixp 8892

This theorem is referenced by:  elptr  23695  ptbasin  23699  ptbasfi  23703  ptrecube  38154