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Theorem ixxval 12431
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 4847 . . . 4 (𝑥 = 𝐴 → (𝑥𝑅𝑧𝐴𝑅𝑧))
21anbi1d 624 . . 3 (𝑥 = 𝐴 → ((𝑥𝑅𝑧𝑧𝑆𝑦) ↔ (𝐴𝑅𝑧𝑧𝑆𝑦)))
32rabbidv 3374 . 2 (𝑥 = 𝐴 → {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)} = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝑦)})
4 breq2 4848 . . . 4 (𝑦 = 𝐵 → (𝑧𝑆𝑦𝑧𝑆𝐵))
54anbi2d 623 . . 3 (𝑦 = 𝐵 → ((𝐴𝑅𝑧𝑧𝑆𝑦) ↔ (𝐴𝑅𝑧𝑧𝑆𝐵)))
65rabbidv 3374 . 2 (𝑦 = 𝐵 → {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝑦)} = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)})
7 ixx.1 . 2 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})
8 xrex 12070 . . 3 * ∈ V
98rabex 5008 . 2 {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)} ∈ V
103, 6, 7, 9ovmpt2 7031 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)})
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  *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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