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Theorem lern 18765

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 13280	. . . 4 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ≤ 𝑥)
2		lerel 11373	. . . . 5 ⊢ Rel ≤
3	2	relelrni 5931	. . . 4 ⊢ (𝑥 ≤ 𝑥 → 𝑥 ∈ ran ≤ )
4	1, 3	syl 18	. . 3 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ∈ ran ≤ )
5	4	ssriv 3935	. 2 ⊢ ℝ^* ⊆ ran ≤
6		lerelxr 11372	. . . 4 ⊢ ≤ ⊆ (ℝ^* × ℝ^*)
7	6	rnssi 5922	. . 3 ⊢ ran ≤ ⊆ ran (ℝ^* × ℝ^*)
8		rnxpss 6164	. . 3 ⊢ ran (ℝ^* × ℝ^) ⊆ ℝ^
9	7, 8	sstri 3940	. 2 ⊢ ran ≤ ⊆ ℝ^*
10	5, 9	eqssi 3947	1 ⊢ ℝ^* = ran ≤

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 ∈ wcel 2145 class class class wbr 5103 × cxp 5649 ran crn 5652 ℝ^*cxr 11342 ≤ cle 11344

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-pre-lttri 11274 ax-pre-lttrn 11275

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 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 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 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-er 8717 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349

This theorem is used by: lefld 18766 cnvordtrestixx 34545 xrge0iifhmeo 34568