ixxf - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ixxf		Structured version Visualization version GIF version

Theorem ixxf 13417

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 13052	. . . 4 ⊢ ℝ^* ∈ V
2		ssrab2 4103	. . . 4 ⊢ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)} ⊆ ℝ^*
3	1, 2	elpwi2 5353	. . 3 ⊢ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)} ∈ 𝒫 ℝ^*
4	3	rgen2w 3072	. 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 8109	. 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 1537 ∈ wcel 2108 ∀wral 3067 {crab 3443 Vcvv 3488 𝒫 cpw 4622 class class class wbr 5166 × cxp 5698 ⟶wf 6569 ∈ cmpo 7450 ℝ^*cxr 11323

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-fv 6581 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031 df-xr 11328

This theorem is referenced by: ixxex 13418 ixxssxr 13419 elixx3g 13420 ndmioo 13434 iccf 13508 iocpnfordt 23244 icomnfordt 23245 tpr2rico 33858 icoreresf 37318 icoreelrn 37327 relowlpssretop 37330 dmico 45483

W3C validator