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 18645

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 13175	. . . 4 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ≤ 𝑥)
2		lerel 11272	. . . . 5 ⊢ Rel ≤
3	2	releldmi 5939	. . . 4 ⊢ (𝑥 ≤ 𝑥 → 𝑥 ∈ dom ≤ )
4	1, 3	syl 18	. . 3 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ∈ dom ≤ )
5	4	ssriv 3949	. 2 ⊢ ℝ^* ⊆ dom ≤
6		lerelxr 11271	. . . 4 ⊢ ≤ ⊆ (ℝ^* × ℝ^*)
7		dmss 5893	. . . 4 ⊢ ( ≤ ⊆ (ℝ^* × ℝ^) → dom ≤ ⊆ dom (ℝ^ × ℝ^*))
8	6, 7	ax-mp 5	. . 3 ⊢ dom ≤ ⊆ dom (ℝ^* × ℝ^*)
9		dmxpss 6170	. . 3 ⊢ dom (ℝ^* × ℝ^) ⊆ ℝ^
10	8, 9	sstri 3954	. 2 ⊢ dom ≤ ⊆ ℝ^*
11	5, 10	eqssi 3961	1 ⊢ ℝ^* = dom ≤

Colors of variables: wff setvar class

Syntax hints: = wceq 1567 ∈ wcel 2149 ⊆ wss 3913 class class class wbr 5113 × cxp 5660 dom cdm 5662 ℝ^*cxr 11241 ≤ cle 11243

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11155 ax-resscn 11156 ax-pre-lttri 11173 ax-pre-lttrn 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-br 5114 df-opab 5178 df-mpt 5197 df-id 5557 df-po 5570 df-so 5571 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-er 8693 df-en 8943 df-dom 8944 df-sdom 8945 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248

This theorem is referenced by: lefld 18647 letsr 18648 letopon 23330 leordtval2 23337 leordtval 23338 iccordt 23339 ordtrestixx 23347 icopnfhmeo 25070 iccpnfhmeo 25072 xrhmeo 25073 xrmulc1cn 34264 xrge0iifhmeo 34270