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Mirrors > Home > MPE Home > Th. List > itg2lr | Structured version Visualization version GIF version |
Description: Sufficient condition for elementhood in the set 𝐿. (Contributed by Mario Carneiro, 28-Jun-2014.) |
Ref | Expression |
---|---|
itg2val.1 | ⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} |
Ref | Expression |
---|---|
itg2lr | ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (∫1‘𝐺) ∈ 𝐿) |
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) |
Colors of variables: wff setvar class |
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 |
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