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Theorem ledm 18622

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 13153	. . . 4 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ≤ 𝑥)
2		lerel 11246	. . . . 5 ⊢ Rel ≤
3	2	releldmi 5924	. . . 4 ⊢ (𝑥 ≤ 𝑥 → 𝑥 ∈ dom ≤ )
4	1, 3	syl 17	. . 3 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ∈ dom ≤ )
5	4	ssriv 3940	. 2 ⊢ ℝ^* ⊆ dom ≤
6		lerelxr 11245	. . . 4 ⊢ ≤ ⊆ (ℝ^* × ℝ^*)
7		dmss 5878	. . . 4 ⊢ ( ≤ ⊆ (ℝ^* × ℝ^) → dom ≤ ⊆ dom (ℝ^ × ℝ^*))
8	6, 7	ax-mp 5	. . 3 ⊢ dom ≤ ⊆ dom (ℝ^* × ℝ^*)
9		dmxpss 6157	. . 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 1560 ∈ wcel 2142 ⊆ wss 3904 class class class wbr 5100 × cxp 5645 dom cdm 5647 ℝ^*cxr 11215 ≤ cle 11217

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-pre-lttri 11147 ax-pre-lttrn 11148

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5542 df-po 5555 df-so 5556 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-er 8678 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222

This theorem is referenced by: lefld 18624 letsr 18625 letopon 23265 leordtval2 23272 leordtval 23273 iccordt 23274 ordtrestixx 23282 icopnfhmeo 25005 iccpnfhmeo 25007 xrhmeo 25008 xrmulc1cn 34227 xrge0iifhmeo 34233