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Mirrors > Home > MPE Home > Th. List > relttrcl | Structured version Visualization version GIF version |
Description: The transitive closure of a class is a relation. (Contributed by Scott Fenton, 17-Oct-2024.) |
Ref | Expression |
---|---|
relttrcl | ⊢ Rel t++𝑅 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ttrcl 9746 | . 2 ⊢ t++𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑅(𝑓‘suc 𝑚))} | |
2 | 1 | relopabi 5835 | 1 ⊢ Rel t++𝑅 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∃wex 1776 ∀wral 3059 ∃wrex 3068 ∖ cdif 3960 ∅c0 4339 class class class wbr 5148 Rel wrel 5694 suc csuc 6388 Fn wfn 6558 ‘cfv 6563 ωcom 7887 1oc1o 8498 t++cttrcl 9745 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-opab 5211 df-xp 5695 df-rel 5696 df-ttrcl 9746 |
This theorem is referenced by: brttrcl 9751 ttrclss 9758 ttrclexg 9761 |
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