Theorem itg2l 25925

Description: Elementhood in the set 𝐿 of lower sums of the integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Hypothesis

Ref	Expression
itg2val.1	⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}

Assertion

Ref	Expression
itg2l	⊢ (𝐴 ∈ 𝐿 ↔ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔)))

Distinct variable groups:   𝑥,𝑔,𝐴   𝑔,𝐹,𝑥

Allowed substitution hints:   𝐿(𝑥, 𝑔)

Proof of Theorem itg2l

Step	Hyp	Ref	Expression
1		itg2val.1	. . 3 ⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}
2	1	eleq2i 2858	. 2 ⊢ (𝐴 ∈ 𝐿 ↔ 𝐴 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))})
3		simpr 490	. . . . 5 ⊢ ((𝑔 ∘r ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔)) → 𝐴 = (∫1‘𝑔))
4		fvex 6901	. . . . 5 ⊢ (∫1‘𝑔) ∈ V
5	3, 4	eqeltrdi 2874	. . . 4 ⊢ ((𝑔 ∘r ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔)) → 𝐴 ∈ V)
6	5	rexlimivw 3165	. . 3 ⊢ (∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔)) → 𝐴 ∈ V)
7		eqeq1 2770	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 = (∫1‘𝑔) ↔ 𝐴 = (∫1‘𝑔)))
8	7	anbi2d 642	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) ↔ (𝑔 ∘r ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔))))
9	8	rexbidv 3192	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) ↔ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔))))
10	6, 9	elab3 3648	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} ↔ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔)))
11	2, 10	bitri 278	1 ⊢ (𝐴 ∈ 𝐿 ↔ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {cab 2744  ∃wrex 3092  Vcvv 3458   class class class wbr 5114  dom cdm 5666  ‘cfv 6543   ∘_r cofr 7686   ≤ cle 11262  ∫₁citg1 25811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-nul 5274

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-rex 3093  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-sn 4595  df-pr 4597  df-uni 4878  df-iota 6499  df-fv 6551

This theorem is used by:  itg2lr  25926