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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 12867 | . . . 4 ⊢ (𝑥 ∈ ℝ* → 𝑥 ≤ 𝑥) | |
2 | lerel 11023 | . . . . 5 ⊢ Rel ≤ | |
3 | 2 | releldmi 5854 | . . . 4 ⊢ (𝑥 ≤ 𝑥 → 𝑥 ∈ dom ≤ ) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝑥 ∈ ℝ* → 𝑥 ∈ dom ≤ ) |
5 | 4 | ssriv 3929 | . 2 ⊢ ℝ* ⊆ dom ≤ |
6 | lerelxr 11022 | . . . 4 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
7 | dmss 5808 | . . . 4 ⊢ ( ≤ ⊆ (ℝ* × ℝ*) → dom ≤ ⊆ dom (ℝ* × ℝ*)) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ dom ≤ ⊆ dom (ℝ* × ℝ*) |
9 | dmxpss 6071 | . . 3 ⊢ dom (ℝ* × ℝ*) ⊆ ℝ* | |
10 | 8, 9 | sstri 3934 | . 2 ⊢ dom ≤ ⊆ ℝ* |
11 | 5, 10 | eqssi 3941 | 1 ⊢ ℝ* = dom ≤ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2109 ⊆ wss 3891 class class class wbr 5078 × cxp 5586 dom cdm 5588 ℝ*cxr 10992 ≤ cle 10994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-pre-lttri 10929 ax-pre-lttrn 10930 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-po 5502 df-so 5503 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 |
This theorem is referenced by: lefld 18291 letsr 18292 letopon 22337 leordtval2 22344 leordtval 22345 iccordt 22346 ordtrestixx 22354 icopnfhmeo 24087 iccpnfhmeo 24089 xrhmeo 24090 xrmulc1cn 31859 xrge0iifhmeo 31865 |
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