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Theorem ixxval 13306

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 5088	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑧 ↔ 𝐴𝑅𝑧))
2	1	anbi1d 632	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦) ↔ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝑦)))
3	2	rabbidv 3396	. 2 ⊢ (𝑥 = 𝐴 → {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)} = {𝑧 ∈ ℝ^* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝑦)})
4		breq2 5089	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑧𝑆𝑦 ↔ 𝑧𝑆𝐵))
5	4	anbi2d 631	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝑅𝑧 ∧ 𝑧𝑆𝑦) ↔ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)))
6	5	rabbidv 3396	. 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 12937	. . 3 ⊢ ℝ^* ∈ V
9	8	rabex 5280	. 2 ⊢ {𝑧 ∈ ℝ^* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)} ∈ V
10	3, 6, 7, 9	ovmpo 7527	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ^ ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 {crab 3389 class class class wbr 5085 (class class class)co 7367 ∈ cmpo 7369 ℝ^*cxr 11178

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-sbc 3729 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6454 df-fun 6500 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-xr 11183

This theorem is referenced by: elixx1 13307 ixxin 13315 iooval 13322 iocval 13335 icoval 13336 iccval 13337

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