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Theorem ledm 18668

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 13192	. . . 4 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ≤ 𝑥)
2		lerel 11288	. . . . 5 ⊢ Rel ≤
3	2	releldmi 5940	. . . 4 ⊢ (𝑥 ≤ 𝑥 → 𝑥 ∈ dom ≤ )
4	1, 3	syl 18	. . 3 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ∈ dom ≤ )
5	4	ssriv 3942	. 2 ⊢ ℝ^* ⊆ dom ≤
6		lerelxr 11287	. . . 4 ⊢ ≤ ⊆ (ℝ^* × ℝ^*)
7		dmss 5894	. . . 4 ⊢ ( ≤ ⊆ (ℝ^* × ℝ^) → dom ≤ ⊆ dom (ℝ^ × ℝ^*))
8	6, 7	ax-mp 5	. . 3 ⊢ dom ≤ ⊆ dom (ℝ^* × ℝ^*)
9		dmxpss 6171	. . 3 ⊢ dom (ℝ^* × ℝ^) ⊆ ℝ^
10	8, 9	sstri 3947	. 2 ⊢ dom ≤ ⊆ ℝ^*
11	5, 10	eqssi 3954	1 ⊢ ℝ^* = dom ≤

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 ∈ wcel 2146 ⊆ wss 3906 class class class wbr 5111 × cxp 5661 dom cdm 5663 ℝ^*cxr 11257 ≤ cle 11259

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-pre-lttri 11189 ax-pre-lttrn 11190

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-po 5571 df-so 5572 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264

This theorem is used by: lefld 18670 letsr 18671 letopon 23412 leordtval2 23419 leordtval 23420 iccordt 23421 ordtrestixx 23429 icopnfhmeo 25153 iccpnfhmeo 25155 xrhmeo 25156 xrmulc1cn 34384 xrge0iifhmeo 34390

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