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Theorem lern 17048

Description: The range of ≤ is ℝ^*. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)

**Assertion**
Ref	Expression
lern	⊢ ℝ^* = ran ≤

**Proof of Theorem lern**
Step	Hyp	Ref	Expression
1		xrleid 11859	. . . 4 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ≤ 𝑥)
2		lerel 9981	. . . . 5 ⊢ Rel ≤
3	2	relelrni 5284	. . . 4 ⊢ (𝑥 ≤ 𝑥 → 𝑥 ∈ ran ≤ )
4	1, 3	syl 17	. . 3 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ∈ ran ≤ )
5	4	ssriv 3572	. 2 ⊢ ℝ^* ⊆ ran ≤
6		lerelxr 9980	. . . 4 ⊢ ≤ ⊆ (ℝ^* × ℝ^*)
7		rnss 5275	. . . 4 ⊢ ( ≤ ⊆ (ℝ^* × ℝ^) → ran ≤ ⊆ ran (ℝ^ × ℝ^*))
8	6, 7	ax-mp 5	. . 3 ⊢ ran ≤ ⊆ ran (ℝ^* × ℝ^*)
9		rnxpss 5485	. . 3 ⊢ ran (ℝ^* × ℝ^) ⊆ ℝ^
10	8, 9	sstri 3577	. 2 ⊢ ran ≤ ⊆ ℝ^*
11	5, 10	eqssi 3584	1 ⊢ ℝ^* = ran ≤

Colors of variables: wff setvar class

Syntax hints: = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 class class class wbr 4583 × cxp 5036 ran crn 5039 ℝ^*cxr 9952 ≤ cle 9954

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959

This theorem is referenced by: lefld 17049 cnvordtrestixx 29287 xrge0iifhmeo 29310