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Mirrors > Home > MPE Home > Th. List > reldmopsr | Structured version Visualization version GIF version |
Description: Lemma for ordered power series. (Contributed by Stefan O'Rear, 2-Oct-2015.) |
Ref | Expression |
---|---|
reldmopsr | ⊢ Rel dom ordPwSer |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-opsr 21116 | . 2 ⊢ ordPwSer = (𝑖 ∈ V, 𝑠 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑖 × 𝑖) ↦ ⦋(𝑖 mPwSer 𝑠) / 𝑝⦌(𝑝 sSet 〈(le‘ndx), {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑝) ∧ ([{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑]∃𝑧 ∈ 𝑑 ((𝑥‘𝑧)(lt‘𝑠)(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}〉))) | |
2 | 1 | reldmmpo 7408 | 1 ⊢ Rel dom ordPwSer |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 844 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 {crab 3068 Vcvv 3432 [wsbc 3716 ⦋csb 3832 ⊆ wss 3887 𝒫 cpw 4533 {cpr 4563 〈cop 4567 class class class wbr 5074 {copab 5136 ↦ cmpt 5157 × cxp 5587 ◡ccnv 5588 dom cdm 5589 “ cima 5592 Rel wrel 5594 ‘cfv 6433 (class class class)co 7275 ↑m cmap 8615 Fincfn 8733 ℕcn 11973 ℕ0cn0 12233 sSet csts 16864 ndxcnx 16894 Basecbs 16912 lecple 16969 ltcplt 18026 mPwSer cmps 21107 <bag cltb 21110 ordPwSer copws 21111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-xp 5595 df-rel 5596 df-dm 5599 df-oprab 7279 df-mpo 7280 df-opsr 21116 |
This theorem is referenced by: opsrle 21248 opsrbaslem 21250 opsrbaslemOLD 21251 psr1val 21357 |
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