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 18550

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 13096	. . . 4 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ≤ 𝑥)
2		lerel 11203	. . . . 5 ⊢ Rel ≤
3	2	releldmi 5898	. . . 4 ⊢ (𝑥 ≤ 𝑥 → 𝑥 ∈ dom ≤ )
4	1, 3	syl 17	. . 3 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ∈ dom ≤ )
5	4	ssriv 3926	. 2 ⊢ ℝ^* ⊆ dom ≤
6		lerelxr 11202	. . . 4 ⊢ ≤ ⊆ (ℝ^* × ℝ^*)
7		dmss 5852	. . . 4 ⊢ ( ≤ ⊆ (ℝ^* × ℝ^) → dom ≤ ⊆ dom (ℝ^ × ℝ^*))
8	6, 7	ax-mp 5	. . 3 ⊢ dom ≤ ⊆ dom (ℝ^* × ℝ^*)
9		dmxpss 6130	. . 3 ⊢ dom (ℝ^* × ℝ^) ⊆ ℝ^
10	8, 9	sstri 3932	. 2 ⊢ dom ≤ ⊆ ℝ^*
11	5, 10	eqssi 3939	1 ⊢ ℝ^* = dom ≤

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ∈ wcel 2114 ⊆ wss 3890 class class class wbr 5086 × cxp 5623 dom cdm 5625 ℝ^*cxr 11172 ≤ cle 11174

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 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-pre-lttri 11106 ax-pre-lttrn 11107

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 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5520 df-po 5533 df-so 5534 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179

This theorem is referenced by: lefld 18552 letsr 18553 letopon 23183 leordtval2 23190 leordtval 23191 iccordt 23192 ordtrestixx 23200 icopnfhmeo 24923 iccpnfhmeo 24925 xrhmeo 24926 xrmulc1cn 34093 xrge0iifhmeo 34099