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Mirrors > Home > MPE Home > Th. List > itg2l | Structured version Visualization version GIF version |
Description: Elementhood in the set 𝐿 of lower sums of the integral. (Contributed by Mario Carneiro, 28-Jun-2014.) |
Ref | Expression |
---|---|
itg2val.1 | ⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} |
Ref | Expression |
---|---|
itg2l | ⊢ (𝐴 ∈ 𝐿 ↔ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itg2val.1 | . . 3 ⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} | |
2 | 1 | eleq2i 2693 | . 2 ⊢ (𝐴 ∈ 𝐿 ↔ 𝐴 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}) |
3 | simpr 477 | . . . . 5 ⊢ ((𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔)) → 𝐴 = (∫1‘𝑔)) | |
4 | fvex 6201 | . . . . 5 ⊢ (∫1‘𝑔) ∈ V | |
5 | 3, 4 | syl6eqel 2709 | . . . 4 ⊢ ((𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔)) → 𝐴 ∈ V) |
6 | 5 | rexlimivw 3029 | . . 3 ⊢ (∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔)) → 𝐴 ∈ V) |
7 | eqeq1 2626 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 = (∫1‘𝑔) ↔ 𝐴 = (∫1‘𝑔))) | |
8 | 7 | anbi2d 740 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) ↔ (𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔)))) |
9 | 8 | rexbidv 3052 | . . 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 3358 | . 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 264 | 1 ⊢ (𝐴 ∈ 𝐿 ↔ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {cab 2608 ∃wrex 2913 Vcvv 3200 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-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-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-sn 4178 df-pr 4180 df-uni 4437 df-iota 5851 df-fv 5896 |
This theorem is referenced by: itg2lr 23497 |
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