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Theorem ledm 18635

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 13193	. . . 4 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ≤ 𝑥)
2		lerel 11325	. . . . 5 ⊢ Rel ≤
3	2	releldmi 5959	. . . 4 ⊢ (𝑥 ≤ 𝑥 → 𝑥 ∈ dom ≤ )
4	1, 3	syl 17	. . 3 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ∈ dom ≤ )
5	4	ssriv 3987	. 2 ⊢ ℝ^* ⊆ dom ≤
6		lerelxr 11324	. . . 4 ⊢ ≤ ⊆ (ℝ^* × ℝ^*)
7		dmss 5913	. . . 4 ⊢ ( ≤ ⊆ (ℝ^* × ℝ^) → dom ≤ ⊆ dom (ℝ^ × ℝ^*))
8	6, 7	ax-mp 5	. . 3 ⊢ dom ≤ ⊆ dom (ℝ^* × ℝ^*)
9		dmxpss 6191	. . 3 ⊢ dom (ℝ^* × ℝ^) ⊆ ℝ^
10	8, 9	sstri 3993	. 2 ⊢ dom ≤ ⊆ ℝ^*
11	5, 10	eqssi 4000	1 ⊢ ℝ^* = dom ≤

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 class class class wbr 5143 × cxp 5683 dom cdm 5685 ℝ^*cxr 11294 ≤ cle 11296

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301

This theorem is referenced by: lefld 18637 letsr 18638 letopon 23213 leordtval2 23220 leordtval 23221 iccordt 23222 ordtrestixx 23230 icopnfhmeo 24974 iccpnfhmeo 24976 xrhmeo 24977 xrmulc1cn 33929 xrge0iifhmeo 33935