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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 2622 | . . 3 ⊢ (∫1‘𝐺) = (∫1‘𝐺) | |
2 | breq1 4656 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 ∘𝑟 ≤ 𝐹 ↔ 𝐺 ∘𝑟 ≤ 𝐹)) | |
3 | fveq2 6191 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (∫1‘𝑔) = (∫1‘𝐺)) | |
4 | 3 | eqeq2d 2632 | . . . . 5 ⊢ (𝑔 = 𝐺 → ((∫1‘𝐺) = (∫1‘𝑔) ↔ (∫1‘𝐺) = (∫1‘𝐺))) |
5 | 2, 4 | anbi12d 747 | . . . 4 ⊢ (𝑔 = 𝐺 → ((𝑔 ∘𝑟 ≤ 𝐹 ∧ (∫1‘𝐺) = (∫1‘𝑔)) ↔ (𝐺 ∘𝑟 ≤ 𝐹 ∧ (∫1‘𝐺) = (∫1‘𝐺)))) |
6 | 5 | rspcev 3309 | . . 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 720 | . 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 23496 | . 2 ⊢ ((∫1‘𝐺) ∈ 𝐿 ↔ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ (∫1‘𝐺) = (∫1‘𝑔))) |
10 | 7, 9 | sylibr 224 | 1 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (∫1‘𝐺) ∈ 𝐿) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {cab 2608 ∃wrex 2913 class class class wbr 4653 dom cdm 5114 ‘cfv 5888 ∘𝑟 cofr 6896 ≤ cle 10075 ∫1citg1 23384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 |
This theorem is referenced by: itg2ub 23500 |
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