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Mirrors > Home > MPE Home > Th. List > lern | Structured version Visualization version GIF version |
Description: The range of ≤ is ℝ*. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
lern | ⊢ ℝ* = ran ≤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrleid 13189 | . . . 4 ⊢ (𝑥 ∈ ℝ* → 𝑥 ≤ 𝑥) | |
2 | lerel 11322 | . . . . 5 ⊢ Rel ≤ | |
3 | 2 | relelrni 5962 | . . . 4 ⊢ (𝑥 ≤ 𝑥 → 𝑥 ∈ ran ≤ ) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝑥 ∈ ℝ* → 𝑥 ∈ ran ≤ ) |
5 | 4 | ssriv 3998 | . 2 ⊢ ℝ* ⊆ ran ≤ |
6 | lerelxr 11321 | . . . 4 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
7 | 6 | rnssi 5953 | . . 3 ⊢ ran ≤ ⊆ ran (ℝ* × ℝ*) |
8 | rnxpss 6193 | . . 3 ⊢ ran (ℝ* × ℝ*) ⊆ ℝ* | |
9 | 7, 8 | sstri 4004 | . 2 ⊢ ran ≤ ⊆ ℝ* |
10 | 5, 9 | eqssi 4011 | 1 ⊢ ℝ* = ran ≤ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2105 class class class wbr 5147 × cxp 5686 ran crn 5689 ℝ*cxr 11291 ≤ cle 11293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-pre-lttri 11226 ax-pre-lttrn 11227 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 |
This theorem is referenced by: lefld 18649 cnvordtrestixx 33873 xrge0iifhmeo 33896 |
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