Theorem itg2l 25652

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 2821	. 2 ⊢ (𝐴 ∈ 𝐿 ↔ 𝐴 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))})
3		simpr 484	. . . . 5 ⊢ ((𝑔 ∘r ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔)) → 𝐴 = (∫1‘𝑔))
4		fvex 6904	. . . . 5 ⊢ (∫1‘𝑔) ∈ V
5	3, 4	eqeltrdi 2837	. . . 4 ⊢ ((𝑔 ∘r ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔)) → 𝐴 ∈ V)
6	5	rexlimivw 3147	. . 3 ⊢ (∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔)) → 𝐴 ∈ V)
7		eqeq1 2732	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 = (∫1‘𝑔) ↔ 𝐴 = (∫1‘𝑔)))
8	7	anbi2d 629	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) ↔ (𝑔 ∘r ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔))))
9	8	rexbidv 3174	. . 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 3674	. 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 275	1 ⊢ (𝐴 ∈ 𝐿 ↔ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔)))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {cab 2705  ∃wrex 3066  Vcvv 3470   class class class wbr 5142  dom cdm 5672  ‘cfv 6542   ∘_r cofr 7678   ≤ cle 11273  ∫₁citg1 25537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-nul 5300

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-rex 3067  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-sn 4625  df-pr 4627  df-uni 4904  df-iota 6494  df-fv 6550

This theorem is referenced by:  itg2lr  25653