Theorem elptr 23613

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 1212	. . . 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 1148	. . . 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 7198	. . 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 1213	. . . 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 4074	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝐴 ∖ 𝑤) = (𝐴 ∖ 𝑊))
6	5	raleqdv 3319	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∀𝑦 ∈ (𝐴 ∖ 𝑤)(𝐺‘𝑦) = ∪ (𝐹‘𝑦) ↔ ∀𝑦 ∈ (𝐴 ∖ 𝑊)(𝐺‘𝑦) = ∪ (𝐹‘𝑦)))
7	6	rspcev 3581	. . . . 5 ⊢ ((𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑊)(𝐺‘𝑦) = ∪ (𝐹‘𝑦)) → ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(𝐺‘𝑦) = ∪ (𝐹‘𝑦))
8	7	3ad2ant3 1147	. . . 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 1140	. . 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 6862	. . . . . . 7 ⊢ (ℎ = 𝐺 → (ℎ‘𝑦) = (𝐺‘𝑦))
11	10	eqcomd 2767	. . . . . 6 ⊢ (ℎ = 𝐺 → (𝐺‘𝑦) = (ℎ‘𝑦))
12	11	ixpeq2dv 8891	. . . . 5 ⊢ (ℎ = 𝐺 → X𝑦 ∈ 𝐴 (𝐺‘𝑦) = X𝑦 ∈ 𝐴 (ℎ‘𝑦))
13	12	biantrud 539	. . . 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 6608	. . . . 5 ⊢ (ℎ = 𝐺 → (ℎ Fn 𝐴 ↔ 𝐺 Fn 𝐴))
15	10	eleq1d 2846	. . . . . 6 ⊢ (ℎ = 𝐺 → ((ℎ‘𝑦) ∈ (𝐹‘𝑦) ↔ (𝐺‘𝑦) ∈ (𝐹‘𝑦)))
16	15	ralbidv 3184	. . . . 5 ⊢ (ℎ = 𝐺 → (∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) ∈ (𝐹‘𝑦)))
17	10	eqeq1d 2763	. . . . . 6 ⊢ (ℎ = 𝐺 → ((ℎ‘𝑦) = ∪ (𝐹‘𝑦) ↔ (𝐺‘𝑦) = ∪ (𝐹‘𝑦)))
18	17	rexralbidv 3227	. . . . 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 1456	. . . 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 283	. . 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 3557	. 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 23612	. 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 236	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 1097   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {cab 2739  ∀wral 3075  ∃wrex 3085  Vcvv 3453   ∖ cdif 3901  ∪ cuni 4864   Fn wfn 6512  ‘cfv 6517  Xcixp 8875  Fincfn 8923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ixp 8876

This theorem is referenced by:  elptr2  23614