Theorem itg2lr 23303

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 2610	. . 3 ⊢ (∫1‘𝐺) = (∫1‘𝐺)
2		breq1 4586	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 ∘𝑟 ≤ 𝐹 ↔ 𝐺 ∘𝑟 ≤ 𝐹))
3		fveq2 6103	. . . . . 6 ⊢ (𝑔 = 𝐺 → (∫1‘𝑔) = (∫1‘𝐺))
4	3	eqeq2d 2620	. . . . 5 ⊢ (𝑔 = 𝐺 → ((∫1‘𝐺) = (∫1‘𝑔) ↔ (∫1‘𝐺) = (∫1‘𝐺)))
5	2, 4	anbi12d 743	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑔 ∘𝑟 ≤ 𝐹 ∧ (∫1‘𝐺) = (∫1‘𝑔)) ↔ (𝐺 ∘𝑟 ≤ 𝐹 ∧ (∫1‘𝐺) = (∫1‘𝐺))))
6	5	rspcev 3282	. . 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 716	. 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 23302	. 2 ⊢ ((∫1‘𝐺) ∈ 𝐿 ↔ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ (∫1‘𝐺) = (∫1‘𝑔)))
10	7, 9	sylibr 223	1 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (∫1‘𝐺) ∈ 𝐿)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897   class class class wbr 4583  dom cdm 5038  ‘cfv 5804   ∘_𝑟 cofr 6794   ≤ cle 9954  ∫₁citg1 23190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812

This theorem is referenced by:  itg2ub  23306