ixxf - Metamath Proof Explorer

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Theorem ixxf 13272

Description: The set of intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)

**Hypothesis**
Ref	Expression
ixx.1	⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})

**Assertion**
Ref	Expression
ixxf	⊢ 𝑂:(ℝ^* × ℝ^)⟶𝒫 ℝ^

Distinct variable groups: 𝑥,𝑦,𝑧,𝑅 𝑥,𝑆,𝑦,𝑧 Allowed substitution hints: 𝑂(𝑥,𝑦,𝑧)

**Proof of Theorem ixxf**
Step	Hyp	Ref	Expression
1		xrex 12901	. . . 4 ⊢ ℝ^* ∈ V
2		ssrab2 4021	. . . 4 ⊢ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)} ⊆ ℝ^*
3	1, 2	elpwi2 5270	. . 3 ⊢ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)} ∈ 𝒫 ℝ^*
4	3	rgen2w 3057	. 2 ⊢ ∀𝑥 ∈ ℝ^* ∀𝑦 ∈ ℝ^* {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)} ∈ 𝒫 ℝ^*
5		ixx.1	. . 3 ⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
6	5	fmpo 8012	. 2 ⊢ (∀𝑥 ∈ ℝ^* ∀𝑦 ∈ ℝ^* {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)} ∈ 𝒫 ℝ^* ↔ 𝑂:(ℝ^* × ℝ^)⟶𝒫 ℝ^)
7	4, 6	mpbi 230	1 ⊢ 𝑂:(ℝ^* × ℝ^)⟶𝒫 ℝ^

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 {crab 3390 Vcvv 3430 𝒫 cpw 4542 class class class wbr 5086 × cxp 5620 ⟶wf 6486 ∈ cmpo 7360 ℝ^*cxr 11166

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 2709 ax-sep 5231 ax-nul 5241 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-fv 6498 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-xr 11171

This theorem is referenced by: ixxex 13273 ixxssxr 13274 elixx3g 13275 ndmioo 13289 iccf 13365 iocpnfordt 23158 icomnfordt 23159 tpr2rico 34062 icoreresf 37664 icoreelrn 37673 relowlpssretop 37676 dmico 45997

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