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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 21447 | . 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 7537 | 1 ⊢ Rel dom ordPwSer |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 {crab 3433 Vcvv 3475 [wsbc 3775 ⦋csb 3891 ⊆ wss 3946 𝒫 cpw 4600 {cpr 4628 〈cop 4632 class class class wbr 5146 {copab 5208 ↦ cmpt 5229 × cxp 5672 ◡ccnv 5673 dom cdm 5674 “ cima 5677 Rel wrel 5679 ‘cfv 6539 (class class class)co 7403 ↑m cmap 8815 Fincfn 8934 ℕcn 12207 ℕ0cn0 12467 sSet csts 17091 ndxcnx 17121 Basecbs 17139 lecple 17199 ltcplt 18256 mPwSer cmps 21438 <bag cltb 21441 ordPwSer copws 21442 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-sn 4627 df-pr 4629 df-op 4633 df-br 5147 df-opab 5209 df-xp 5680 df-rel 5681 df-dm 5684 df-oprab 7407 df-mpo 7408 df-opsr 21447 |
This theorem is referenced by: opsrle 21583 opsrbaslem 21585 opsrbaslemOLD 21586 psr1val 21691 |
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