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Theorem ixxval 13314

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^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})

**Assertion**
Ref	Expression
ixxval	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ^ ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)})

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**
Step	Hyp	Ref	Expression
1		breq1 5110	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑧 ↔ 𝐴𝑅𝑧))
2	1	anbi1d 631	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦) ↔ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝑦)))
3	2	rabbidv 3413	. 2 ⊢ (𝑥 = 𝐴 → {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)} = {𝑧 ∈ ℝ^* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝑦)})
4		breq2 5111	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑧𝑆𝑦 ↔ 𝑧𝑆𝐵))
5	4	anbi2d 630	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝑅𝑧 ∧ 𝑧𝑆𝑦) ↔ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)))
6	5	rabbidv 3413	. 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 12946	. . 3 ⊢ ℝ^* ∈ V
9	8	rabex 5294	. 2 ⊢ {𝑧 ∈ ℝ^* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)} ∈ V
10	3, 6, 7, 9	ovmpo 7549	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ^ ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {crab 3405 class class class wbr 5107 (class class class)co 7387 ∈ cmpo 7389 ℝ^*cxr 11207

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-sbc 3754 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6464 df-fun 6513 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-xr 11212

This theorem is referenced by: elixx1 13315 ixxin 13323 iooval 13330 iocval 13343 icoval 13344 iccval 13345

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