Theorem itg2lr 25715

Description: Sufficient condition for elementhood in the set 𝐿. (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
itg2lr	⊢ ((𝐺 ∈ dom ∫1 ∧ 𝐺 ∘r ≤ 𝐹) → (∫1‘𝐺) ∈ 𝐿)

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

Allowed substitution hints:   𝐿(𝑥,𝑔)

Proof of Theorem itg2lr

Step	Hyp	Ref	Expression
1		eqid 2739	. . 3 ⊢ (∫1‘𝐺) = (∫1‘𝐺)
2		breq1 5075	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 ∘r ≤ 𝐹 ↔ 𝐺 ∘r ≤ 𝐹))
3		fveq2 6827	. . . . . 6 ⊢ (𝑔 = 𝐺 → (∫1‘𝑔) = (∫1‘𝐺))
4	3	eqeq2d 2750	. . . . 5 ⊢ (𝑔 = 𝐺 → ((∫1‘𝐺) = (∫1‘𝑔) ↔ (∫1‘𝐺) = (∫1‘𝐺)))
5	2, 4	anbi12d 638	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑔 ∘r ≤ 𝐹 ∧ (∫1‘𝐺) = (∫1‘𝑔)) ↔ (𝐺 ∘r ≤ 𝐹 ∧ (∫1‘𝐺) = (∫1‘𝐺))))
6	5	rspcev 3560	. . 3 ⊢ ((𝐺 ∈ dom ∫1 ∧ (𝐺 ∘r ≤ 𝐹 ∧ (∫1‘𝐺) = (∫1‘𝐺))) → ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ (∫1‘𝐺) = (∫1‘𝑔)))
7	1, 6	mpanr2 710	. 2 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝐺 ∘r ≤ 𝐹) → ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ (∫1‘𝐺) = (∫1‘𝑔)))
8		itg2val.1	. . 3 ⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}
9	8	itg2l 25714	. 2 ⊢ ((∫1‘𝐺) ∈ 𝐿 ↔ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ (∫1‘𝐺) = (∫1‘𝑔)))
10	7, 9	sylibr 235	1 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝐺 ∘r ≤ 𝐹) → (∫1‘𝐺) ∈ 𝐿)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2717  ∃wrex 3063   class class class wbr 5072  dom cdm 5618  ‘cfv 6485   ∘_r cofr 7619   ≤ cle 11171  ∫₁citg1 25600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-nul 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493

This theorem is referenced by:  itg2ub  25718