Theorem elptr 22273

Description: A basic open set in the 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
elptr	⊢ ((𝐴 ∈ 𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) ∈ (𝐹‘𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑊)(𝐺‘𝑦) = ∪ (𝐹‘𝑦))) → X𝑦 ∈ 𝐴 (𝐺‘𝑦) ∈ 𝐵)

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

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

Proof of Theorem elptr

Dummy variables ℎ 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2l 1196	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) ∈ (𝐹‘𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑊)(𝐺‘𝑦) = ∪ (𝐹‘𝑦))) → 𝐺 Fn 𝐴)
2		simp1 1133	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) ∈ (𝐹‘𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑊)(𝐺‘𝑦) = ∪ (𝐹‘𝑦))) → 𝐴 ∈ 𝑉)
3	1, 2	fnexd 6972	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) ∈ (𝐹‘𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑊)(𝐺‘𝑦) = ∪ (𝐹‘𝑦))) → 𝐺 ∈ V)
4		simp2r 1197	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) ∈ (𝐹‘𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑊)(𝐺‘𝑦) = ∪ (𝐹‘𝑦))) → ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) ∈ (𝐹‘𝑦))
5		difeq2 4022	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝐴 ∖ 𝑤) = (𝐴 ∖ 𝑊))
6	5	raleqdv 3329	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∀𝑦 ∈ (𝐴 ∖ 𝑤)(𝐺‘𝑦) = ∪ (𝐹‘𝑦) ↔ ∀𝑦 ∈ (𝐴 ∖ 𝑊)(𝐺‘𝑦) = ∪ (𝐹‘𝑦)))
7	6	rspcev 3541	. . . . 5 ⊢ ((𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑊)(𝐺‘𝑦) = ∪ (𝐹‘𝑦)) → ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(𝐺‘𝑦) = ∪ (𝐹‘𝑦))
8	7	3ad2ant3 1132	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) ∈ (𝐹‘𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑊)(𝐺‘𝑦) = ∪ (𝐹‘𝑦))) → ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(𝐺‘𝑦) = ∪ (𝐹‘𝑦))
9	1, 4, 8	3jca 1125	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) ∈ (𝐹‘𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑊)(𝐺‘𝑦) = ∪ (𝐹‘𝑦))) → (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(𝐺‘𝑦) = ∪ (𝐹‘𝑦)))
10		fveq1 6657	. . . . . . 7 ⊢ (ℎ = 𝐺 → (ℎ‘𝑦) = (𝐺‘𝑦))
11	10	eqcomd 2764	. . . . . 6 ⊢ (ℎ = 𝐺 → (𝐺‘𝑦) = (ℎ‘𝑦))
12	11	ixpeq2dv 8495	. . . . 5 ⊢ (ℎ = 𝐺 → X𝑦 ∈ 𝐴 (𝐺‘𝑦) = X𝑦 ∈ 𝐴 (ℎ‘𝑦))
13	12	biantrud 535	. . . 4 ⊢ (ℎ = 𝐺 → ((ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)) ↔ ((ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)) ∧ X𝑦 ∈ 𝐴 (𝐺‘𝑦) = X𝑦 ∈ 𝐴 (ℎ‘𝑦))))
14		fneq1 6425	. . . . 5 ⊢ (ℎ = 𝐺 → (ℎ Fn 𝐴 ↔ 𝐺 Fn 𝐴))
15	10	eleq1d 2836	. . . . . 6 ⊢ (ℎ = 𝐺 → ((ℎ‘𝑦) ∈ (𝐹‘𝑦) ↔ (𝐺‘𝑦) ∈ (𝐹‘𝑦)))
16	15	ralbidv 3126	. . . . 5 ⊢ (ℎ = 𝐺 → (∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) ∈ (𝐹‘𝑦)))
17	10	eqeq1d 2760	. . . . . 6 ⊢ (ℎ = 𝐺 → ((ℎ‘𝑦) = ∪ (𝐹‘𝑦) ↔ (𝐺‘𝑦) = ∪ (𝐹‘𝑦)))
18	17	rexralbidv 3225	. . . . 5 ⊢ (ℎ = 𝐺 → (∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦) ↔ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(𝐺‘𝑦) = ∪ (𝐹‘𝑦)))
19	14, 16, 18	3anbi123d 1433	. . . 4 ⊢ (ℎ = 𝐺 → ((ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)) ↔ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(𝐺‘𝑦) = ∪ (𝐹‘𝑦))))
20	13, 19	bitr3d 284	. . 3 ⊢ (ℎ = 𝐺 → (((ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)) ∧ X𝑦 ∈ 𝐴 (𝐺‘𝑦) = X𝑦 ∈ 𝐴 (ℎ‘𝑦)) ↔ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(𝐺‘𝑦) = ∪ (𝐹‘𝑦))))
21	3, 9, 20	spcedv 3517	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) ∈ (𝐹‘𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑊)(𝐺‘𝑦) = ∪ (𝐹‘𝑦))) → ∃ℎ((ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)) ∧ X𝑦 ∈ 𝐴 (𝐺‘𝑦) = X𝑦 ∈ 𝐴 (ℎ‘𝑦)))
22		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))}
23	22	elpt 22272	. 2 ⊢ (X𝑦 ∈ 𝐴 (𝐺‘𝑦) ∈ 𝐵 ↔ ∃ℎ((ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)) ∧ X𝑦 ∈ 𝐴 (𝐺‘𝑦) = X𝑦 ∈ 𝐴 (ℎ‘𝑦)))
24	21, 23	sylibr 237	1 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) ∈ (𝐹‘𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑊)(𝐺‘𝑦) = ∪ (𝐹‘𝑦))) → X𝑦 ∈ 𝐴 (𝐺‘𝑦) ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2735  ∀wral 3070  ∃wrex 3071  Vcvv 3409   ∖ cdif 3855  ∪ cuni 4798   Fn wfn 6330  ‘cfv 6335  Xcixp 8479  Fincfn 8527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ixp 8480

This theorem is referenced by:  elptr2  22274