Theorem itg2l 24332

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 2906	. 2 ⊢ (𝐴 ∈ 𝐿 ↔ 𝐴 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))})
3		simpr 487	. . . . 5 ⊢ ((𝑔 ∘r ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔)) → 𝐴 = (∫1‘𝑔))
4		fvex 6685	. . . . 5 ⊢ (∫1‘𝑔) ∈ V
5	3, 4	eqeltrdi 2923	. . . 4 ⊢ ((𝑔 ∘r ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔)) → 𝐴 ∈ V)
6	5	rexlimivw 3284	. . 3 ⊢ (∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔)) → 𝐴 ∈ V)
7		eqeq1 2827	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 = (∫1‘𝑔) ↔ 𝐴 = (∫1‘𝑔)))
8	7	anbi2d 630	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) ↔ (𝑔 ∘r ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔))))
9	8	rexbidv 3299	. . 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 3676	. 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 277	1 ⊢ (𝐴 ∈ 𝐿 ↔ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔)))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {cab 2801  ∃wrex 3141  Vcvv 3496   class class class wbr 5068  dom cdm 5557  ‘cfv 6357   ∘_r cofr 7410   ≤ cle 10678  ∫₁citg1 24218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-nul 5212

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-sn 4570  df-pr 4572  df-uni 4841  df-iota 6316  df-fv 6365

This theorem is referenced by:  itg2lr  24333