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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 12706 | . . . 4 ⊢ (𝑥 ∈ ℝ* → 𝑥 ≤ 𝑥) | |
2 | lerel 10862 | . . . . 5 ⊢ Rel ≤ | |
3 | 2 | relelrni 5803 | . . . 4 ⊢ (𝑥 ≤ 𝑥 → 𝑥 ∈ ran ≤ ) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝑥 ∈ ℝ* → 𝑥 ∈ ran ≤ ) |
5 | 4 | ssriv 3891 | . 2 ⊢ ℝ* ⊆ ran ≤ |
6 | lerelxr 10861 | . . . 4 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
7 | 6 | rnssi 5794 | . . 3 ⊢ ran ≤ ⊆ ran (ℝ* × ℝ*) |
8 | rnxpss 6015 | . . 3 ⊢ ran (ℝ* × ℝ*) ⊆ ℝ* | |
9 | 7, 8 | sstri 3896 | . 2 ⊢ ran ≤ ⊆ ℝ* |
10 | 5, 9 | eqssi 3903 | 1 ⊢ ℝ* = ran ≤ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2112 class class class wbr 5039 × cxp 5534 ran crn 5537 ℝ*cxr 10831 ≤ cle 10833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-pre-lttri 10768 ax-pre-lttrn 10769 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 |
This theorem is referenced by: lefld 18052 cnvordtrestixx 31531 xrge0iifhmeo 31554 |
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