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 13275

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 12904	. . . 4 ⊢ ℝ^* ∈ V
2		ssrab2 4033	. . . 4 ⊢ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)} ⊆ ℝ^*
3	1, 2	elpwi2 5281	. . 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 8014	. 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 3400 Vcvv 3441 𝒫 cpw 4555 class class class wbr 5099 × cxp 5623 ⟶wf 6489 ∈ cmpo 7362 ℝ^*cxr 11169

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 5242 ax-nul 5252 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087

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 3062 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-oprab 7364 df-mpo 7365 df-1st 7935 df-2nd 7936 df-xr 11174

This theorem is referenced by: ixxex 13276 ixxssxr 13277 elixx3g 13278 ndmioo 13292 iccf 13368 iocpnfordt 23163 icomnfordt 23164 tpr2rico 34050 icoreresf 37528 icoreelrn 37537 relowlpssretop 37540 dmico 45845

W3C validator