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Theorem ixxval 10298

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 4133	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑧 ↔ 𝐴𝑅𝑧))
2	1	anbi1d 469	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦) ↔ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝑦)))
3	2	rabbidv 2810	. 2 ⊢ (𝑥 = 𝐴 → {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)} = {𝑧 ∈ ℝ^* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝑦)})
4		breq2 4134	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑧𝑆𝑦 ↔ 𝑧𝑆𝐵))
5	4	anbi2d 468	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝑅𝑧 ∧ 𝑧𝑆𝑦) ↔ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)))
6	5	rabbidv 2810	. 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 10258	. . 3 ⊢ ℝ^* ∈ V
9	8	rabex 4280	. 2 ⊢ {𝑧 ∈ ℝ^* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)} ∈ V
10	3, 6, 7, 9	ovmpo 6224	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ^ ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)})

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 = wceq 1402 ∈ wcel 2209 {crab 2532 class class class wbr 4130 (class class class)co 6085 ∈ cmpo 6087 ℝ^*cxr 8359

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-cnex 8270 ax-resscn 8271

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-br 4131 df-opab 4193 df-id 4438 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-iota 5337 df-fun 5379 df-fv 5385 df-ov 6088 df-oprab 6089 df-mpo 6090 df-pnf 8362 df-mnf 8363 df-xr 8364

This theorem is used by: elixx1 10299 iooval 10310 iocval 10320 icoval 10321 iccval 10322

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