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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 20719 | . 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 7294 | 1 ⊢ Rel dom ordPwSer |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 846 = wceq 1542 ∈ wcel 2113 ∀wral 3053 ∃wrex 3054 {crab 3057 Vcvv 3397 [wsbc 3679 ⦋csb 3788 ⊆ wss 3841 𝒫 cpw 4485 {cpr 4515 〈cop 4519 class class class wbr 5027 {copab 5089 ↦ cmpt 5107 × cxp 5517 ◡ccnv 5518 dom cdm 5519 “ cima 5522 Rel wrel 5524 ‘cfv 6333 (class class class)co 7164 ↑m cmap 8430 Fincfn 8548 ℕcn 11709 ℕ0cn0 11969 ndxcnx 16576 sSet csts 16577 Basecbs 16579 lecple 16668 ltcplt 17660 mPwSer cmps 20710 <bag cltb 20713 ordPwSer copws 20714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-v 3399 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-br 5028 df-opab 5090 df-xp 5525 df-rel 5526 df-dm 5529 df-oprab 7168 df-mpo 7169 df-opsr 20719 |
This theorem is referenced by: opsrle 20851 opsrbaslem 20853 psr1val 20954 |
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