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Mirrors > Home > MPE Home > Th. List > ledm | Structured version Visualization version GIF version |
Description: The domain of ≤ is ℝ*. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.) |
Ref | Expression |
---|---|
ledm | ⊢ ℝ* = dom ≤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrleid 13127 | . . . 4 ⊢ (𝑥 ∈ ℝ* → 𝑥 ≤ 𝑥) | |
2 | lerel 11275 | . . . . 5 ⊢ Rel ≤ | |
3 | 2 | releldmi 5937 | . . . 4 ⊢ (𝑥 ≤ 𝑥 → 𝑥 ∈ dom ≤ ) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝑥 ∈ ℝ* → 𝑥 ∈ dom ≤ ) |
5 | 4 | ssriv 3978 | . 2 ⊢ ℝ* ⊆ dom ≤ |
6 | lerelxr 11274 | . . . 4 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
7 | dmss 5892 | . . . 4 ⊢ ( ≤ ⊆ (ℝ* × ℝ*) → dom ≤ ⊆ dom (ℝ* × ℝ*)) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ dom ≤ ⊆ dom (ℝ* × ℝ*) |
9 | dmxpss 6160 | . . 3 ⊢ dom (ℝ* × ℝ*) ⊆ ℝ* | |
10 | 8, 9 | sstri 3983 | . 2 ⊢ dom ≤ ⊆ ℝ* |
11 | 5, 10 | eqssi 3990 | 1 ⊢ ℝ* = dom ≤ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ⊆ wss 3940 class class class wbr 5138 × cxp 5664 dom cdm 5666 ℝ*cxr 11244 ≤ cle 11246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-pre-lttri 11180 ax-pre-lttrn 11181 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 |
This theorem is referenced by: lefld 18547 letsr 18548 letopon 23031 leordtval2 23038 leordtval 23039 iccordt 23040 ordtrestixx 23048 icopnfhmeo 24790 iccpnfhmeo 24792 xrhmeo 24793 xrmulc1cn 33399 xrge0iifhmeo 33405 |
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