ledm - Metamath Proof Explorer

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

Theorem ledm 18525

Description: The domain of ≤ is ℝ^*. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.)

**Assertion**
Ref	Expression
ledm	⊢ ℝ^* = dom ≤

**Proof of Theorem ledm**
Step	Hyp	Ref	Expression
1		xrleid 13077	. . . 4 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ≤ 𝑥)
2		lerel 11208	. . . . 5 ⊢ Rel ≤
3	2	releldmi 5905	. . . 4 ⊢ (𝑥 ≤ 𝑥 → 𝑥 ∈ dom ≤ )
4	1, 3	syl 17	. . 3 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ∈ dom ≤ )
5	4	ssriv 3939	. 2 ⊢ ℝ^* ⊆ dom ≤
6		lerelxr 11207	. . . 4 ⊢ ≤ ⊆ (ℝ^* × ℝ^*)
7		dmss 5859	. . . 4 ⊢ ( ≤ ⊆ (ℝ^* × ℝ^) → dom ≤ ⊆ dom (ℝ^ × ℝ^*))
8	6, 7	ax-mp 5	. . 3 ⊢ dom ≤ ⊆ dom (ℝ^* × ℝ^*)
9		dmxpss 6137	. . 3 ⊢ dom (ℝ^* × ℝ^) ⊆ ℝ^
10	8, 9	sstri 3945	. 2 ⊢ dom ≤ ⊆ ℝ^*
11	5, 10	eqssi 3952	1 ⊢ ℝ^* = dom ≤

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ∈ wcel 2114 ⊆ wss 3903 class class class wbr 5100 × cxp 5630 dom cdm 5632 ℝ^*cxr 11177 ≤ cle 11179

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 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-pre-lttri 11112 ax-pre-lttrn 11113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 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-nel 3038 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-po 5540 df-so 5541 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184

This theorem is referenced by: lefld 18527 letsr 18528 letopon 23161 leordtval2 23168 leordtval 23169 iccordt 23170 ordtrestixx 23178 icopnfhmeo 24909 iccpnfhmeo 24911 xrhmeo 24912 xrmulc1cn 34108 xrge0iifhmeo 34114