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Mirrors > Home > MPE Home > Th. List > ixxval | Structured version Visualization version GIF version |
Description: Value of the interval function. (Contributed by Mario Carneiro, 3-Nov-2013.) |
Ref | Expression |
---|---|
ixx.1 | ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) |
Ref | Expression |
---|---|
ixxval | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 4847 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑧 ↔ 𝐴𝑅𝑧)) | |
2 | 1 | anbi1d 624 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦) ↔ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝑦))) |
3 | 2 | rabbidv 3374 | . 2 ⊢ (𝑥 = 𝐴 → {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)} = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝑦)}) |
4 | breq2 4848 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝑧𝑆𝑦 ↔ 𝑧𝑆𝐵)) | |
5 | 4 | anbi2d 623 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝑅𝑧 ∧ 𝑧𝑆𝑦) ↔ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵))) |
6 | 5 | rabbidv 3374 | . 2 ⊢ (𝑦 = 𝐵 → {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝑦)} = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)}) |
7 | ixx.1 | . 2 ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) | |
8 | xrex 12070 | . . 3 ⊢ ℝ* ∈ V | |
9 | 8 | rabex 5008 | . 2 ⊢ {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)} ∈ V |
10 | 3, 6, 7, 9 | ovmpt2 7031 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 {crab 3094 class class class wbr 4844 (class class class)co 6879 ↦ cmpt2 6881 ℝ*cxr 10363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pr 5098 ax-un 7184 ax-cnex 10281 ax-resscn 10282 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3388 df-sbc 3635 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-br 4845 df-opab 4907 df-id 5221 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-iota 6065 df-fun 6104 df-fv 6110 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-xr 10368 |
This theorem is referenced by: elixx1 12432 ixxin 12440 iooval 12447 iocval 12460 icoval 12461 iccval 12462 |
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