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Theorem lern 18815

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 13320	. . . 4 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ≤ 𝑥)
2		lerel 11412	. . . . 5 ⊢ Rel ≤
3	2	relelrni 5943	. . . 4 ⊢ (𝑥 ≤ 𝑥 → 𝑥 ∈ ran ≤ )
4	1, 3	syl 18	. . 3 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ∈ ran ≤ )
5	4	ssriv 3917	. 2 ⊢ ℝ^* ⊆ ran ≤
6		lerelxr 11411	. . . 4 ⊢ ≤ ⊆ (ℝ^* × ℝ^*)
7	6	rnssi 5934	. . 3 ⊢ ran ≤ ⊆ ran (ℝ^* × ℝ^*)
8		rnxpss 6182	. . 3 ⊢ ran (ℝ^* × ℝ^) ⊆ ℝ^
9	7, 8	sstri 3922	. 2 ⊢ ran ≤ ⊆ ℝ^*
10	5, 9	eqssi 3929	1 ⊢ ℝ^* = ran ≤

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 ∈ wcel 2147 class class class wbr 5108 × cxp 5656 ran crn 5659 ℝ^*cxr 11381 ≤ cle 11383

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-pre-lttri 11313 ax-pre-lttrn 11314

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-po 5566 df-so 5567 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388

This theorem is used by: lefld 18816 cnvordtrestixx 34656 xrge0iifhmeo 34679