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Theorem ixxval 13392

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 5151	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑧 ↔ 𝐴𝑅𝑧))
2	1	anbi1d 631	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦) ↔ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝑦)))
3	2	rabbidv 3441	. 2 ⊢ (𝑥 = 𝐴 → {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)} = {𝑧 ∈ ℝ^* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝑦)})
4		breq2 5152	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑧𝑆𝑦 ↔ 𝑧𝑆𝐵))
5	4	anbi2d 630	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝑅𝑧 ∧ 𝑧𝑆𝑦) ↔ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)))
6	5	rabbidv 3441	. 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 13027	. . 3 ⊢ ℝ^* ∈ V
9	8	rabex 5345	. 2 ⊢ {𝑧 ∈ ℝ^* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)} ∈ V
10	3, 6, 7, 9	ovmpo 7593	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ^ ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 class class class wbr 5148 (class class class)co 7431 ∈ cmpo 7433 ℝ^*cxr 11292

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-xr 11297

This theorem is referenced by: elixx1 13393 ixxin 13401 iooval 13408 iocval 13421 icoval 13422 iccval 13423

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