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Theorem ixxval 13396
Description: Value of the interval function. (Contributed by Mario Carneiro, 3-Nov-2013.)
Hypothesis
Ref Expression
ixx.1 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})
Assertion
Ref Expression
ixxval ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)})
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧
Allowed substitution hints:   𝑂(𝑥,𝑦,𝑧)

Proof of Theorem ixxval
StepHypRef Expression
1 breq1 5145 . . . 4 (𝑥 = 𝐴 → (𝑥𝑅𝑧𝐴𝑅𝑧))
21anbi1d 631 . . 3 (𝑥 = 𝐴 → ((𝑥𝑅𝑧𝑧𝑆𝑦) ↔ (𝐴𝑅𝑧𝑧𝑆𝑦)))
32rabbidv 3443 . 2 (𝑥 = 𝐴 → {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)} = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝑦)})
4 breq2 5146 . . . 4 (𝑦 = 𝐵 → (𝑧𝑆𝑦𝑧𝑆𝐵))
54anbi2d 630 . . 3 (𝑦 = 𝐵 → ((𝐴𝑅𝑧𝑧𝑆𝑦) ↔ (𝐴𝑅𝑧𝑧𝑆𝐵)))
65rabbidv 3443 . 2 (𝑦 = 𝐵 → {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝑦)} = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)})
7 ixx.1 . 2 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})
8 xrex 13030 . . 3 * ∈ V
98rabex 5338 . 2 {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)} ∈ V
103, 6, 7, 9ovmpo 7594 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  {crab 3435   class class class wbr 5142  (class class class)co 7432  cmpo 7434  *cxr 11295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-xr 11300
This theorem is referenced by:  elixx1  13397  ixxin  13405  iooval  13412  iocval  13425  icoval  13426  iccval  13427
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