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Theorem ixxval 13226
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 5106 . . . 4 (𝑥 = 𝐴 → (𝑥𝑅𝑧𝐴𝑅𝑧))
21anbi1d 630 . . 3 (𝑥 = 𝐴 → ((𝑥𝑅𝑧𝑧𝑆𝑦) ↔ (𝐴𝑅𝑧𝑧𝑆𝑦)))
32rabbidv 3413 . 2 (𝑥 = 𝐴 → {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)} = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝑦)})
4 breq2 5107 . . . 4 (𝑦 = 𝐵 → (𝑧𝑆𝑦𝑧𝑆𝐵))
54anbi2d 629 . . 3 (𝑦 = 𝐵 → ((𝐴𝑅𝑧𝑧𝑆𝑦) ↔ (𝐴𝑅𝑧𝑧𝑆𝐵)))
65rabbidv 3413 . 2 (𝑦 = 𝐵 → {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝑦)} = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)})
7 ixx.1 . 2 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})
8 xrex 12866 . . 3 * ∈ V
98rabex 5287 . 2 {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)} ∈ V
103, 6, 7, 9ovmpo 7509 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {crab 3405   class class class wbr 5103  (class class class)co 7351  cmpo 7353  *cxr 11146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664  ax-cnex 11065  ax-resscn 11066
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-xr 11151
This theorem is referenced by:  elixx1  13227  ixxin  13235  iooval  13242  iocval  13255  icoval  13256  iccval  13257
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