Theorem elptr 22724

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 1198	. . . 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 1135	. . . 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 7094	. . 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 1199	. . . 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 4051	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝐴 ∖ 𝑤) = (𝐴 ∖ 𝑊))
6	5	raleqdv 3348	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∀𝑦 ∈ (𝐴 ∖ 𝑤)(𝐺‘𝑦) = ∪ (𝐹‘𝑦) ↔ ∀𝑦 ∈ (𝐴 ∖ 𝑊)(𝐺‘𝑦) = ∪ (𝐹‘𝑦)))
7	6	rspcev 3561	. . . . 5 ⊢ ((𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑊)(𝐺‘𝑦) = ∪ (𝐹‘𝑦)) → ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(𝐺‘𝑦) = ∪ (𝐹‘𝑦))
8	7	3ad2ant3 1134	. . . 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 1127	. . 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 6773	. . . . . . 7 ⊢ (ℎ = 𝐺 → (ℎ‘𝑦) = (𝐺‘𝑦))
11	10	eqcomd 2744	. . . . . 6 ⊢ (ℎ = 𝐺 → (𝐺‘𝑦) = (ℎ‘𝑦))
12	11	ixpeq2dv 8701	. . . . 5 ⊢ (ℎ = 𝐺 → X𝑦 ∈ 𝐴 (𝐺‘𝑦) = X𝑦 ∈ 𝐴 (ℎ‘𝑦))
13	12	biantrud 532	. . . 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 6524	. . . . 5 ⊢ (ℎ = 𝐺 → (ℎ Fn 𝐴 ↔ 𝐺 Fn 𝐴))
15	10	eleq1d 2823	. . . . . 6 ⊢ (ℎ = 𝐺 → ((ℎ‘𝑦) ∈ (𝐹‘𝑦) ↔ (𝐺‘𝑦) ∈ (𝐹‘𝑦)))
16	15	ralbidv 3112	. . . . 5 ⊢ (ℎ = 𝐺 → (∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) ∈ (𝐹‘𝑦)))
17	10	eqeq1d 2740	. . . . . 6 ⊢ (ℎ = 𝐺 → ((ℎ‘𝑦) = ∪ (𝐹‘𝑦) ↔ (𝐺‘𝑦) = ∪ (𝐹‘𝑦)))
18	17	rexralbidv 3230	. . . . 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 1435	. . . 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 280	. . 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 3537	. 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 22723	. 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 233	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 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∖ cdif 3884  ∪ cuni 4839   Fn wfn 6428  ‘cfv 6433  Xcixp 8685  Fincfn 8733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ixp 8686

This theorem is referenced by:  elptr2  22725