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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 33504 | . 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 5689 | 1 ⊢ Rel t++𝑅 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∃wex 1787 ∀wral 3058 ∃wrex 3059 ∖ cdif 3860 ∅c0 4234 class class class wbr 5050 Rel wrel 5553 suc csuc 6212 Fn wfn 6372 ‘cfv 6377 ωcom 7641 1oc1o 8192 t++cttrcl 33503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-11 2158 ax-12 2175 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3067 df-v 3407 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-nul 4235 df-if 4437 df-sn 4539 df-pr 4541 df-op 4545 df-opab 5113 df-xp 5554 df-rel 5555 df-ttrcl 33504 |
This theorem is referenced by: brttrcl 33509 ttrclss 33516 ttrclexg 33519 |
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