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Theorem lern 18774

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 13287	. . . 4 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ≤ 𝑥)
2		lerel 11380	. . . . 5 ⊢ Rel ≤
3	2	relelrni 5932	. . . 4 ⊢ (𝑥 ≤ 𝑥 → 𝑥 ∈ ran ≤ )
4	1, 3	syl 18	. . 3 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ∈ ran ≤ )
5	4	ssriv 3913	. 2 ⊢ ℝ^* ⊆ ran ≤
6		lerelxr 11379	. . . 4 ⊢ ≤ ⊆ (ℝ^* × ℝ^*)
7	6	rnssi 5923	. . 3 ⊢ ran ≤ ⊆ ran (ℝ^* × ℝ^*)
8		rnxpss 6166	. . 3 ⊢ ran (ℝ^* × ℝ^) ⊆ ℝ^
9	7, 8	sstri 3918	. 2 ⊢ ran ≤ ⊆ ℝ^*
10	5, 9	eqssi 3925	1 ⊢ ℝ^* = ran ≤

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1569 ∈ wcel 2144 class class class wbr 5102 × cxp 5649 ran crn 5652 ℝ^*cxr 11349 ≤ cle 11351

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-pre-lttri 11281 ax-pre-lttrn 11282

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5546 df-po 5559 df-so 5560 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-er 8725 df-en 8982 df-dom 8983 df-sdom 8984 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356

This theorem is used by: lefld 18775 cnvordtrestixx 34595 xrge0iifhmeo 34618