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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 12532 | . . . 4 ⊢ (𝑥 ∈ ℝ* → 𝑥 ≤ 𝑥) | |
2 | lerel 10694 | . . . . 5 ⊢ Rel ≤ | |
3 | 2 | releldmi 5782 | . . . 4 ⊢ (𝑥 ≤ 𝑥 → 𝑥 ∈ dom ≤ ) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝑥 ∈ ℝ* → 𝑥 ∈ dom ≤ ) |
5 | 4 | ssriv 3919 | . 2 ⊢ ℝ* ⊆ dom ≤ |
6 | lerelxr 10693 | . . . 4 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
7 | dmss 5735 | . . . 4 ⊢ ( ≤ ⊆ (ℝ* × ℝ*) → dom ≤ ⊆ dom (ℝ* × ℝ*)) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ dom ≤ ⊆ dom (ℝ* × ℝ*) |
9 | dmxpss 5995 | . . 3 ⊢ dom (ℝ* × ℝ*) ⊆ ℝ* | |
10 | 8, 9 | sstri 3924 | . 2 ⊢ dom ≤ ⊆ ℝ* |
11 | 5, 10 | eqssi 3931 | 1 ⊢ ℝ* = dom ≤ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 class class class wbr 5030 × cxp 5517 dom cdm 5519 ℝ*cxr 10663 ≤ cle 10665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 |
This theorem is referenced by: lefld 17828 letsr 17829 letopon 21810 leordtval2 21817 leordtval 21818 iccordt 21819 ordtrestixx 21827 icopnfhmeo 23548 iccpnfhmeo 23550 xrhmeo 23551 xrmulc1cn 31283 xrge0iifhmeo 31289 |
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