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| 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 9749 | . 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 5831 | 1 ⊢ Rel t++𝑅 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∃wex 1778 ∀wral 3060 ∃wrex 3069 ∖ cdif 3947 ∅c0 4332 class class class wbr 5142 Rel wrel 5689 suc csuc 6385 Fn wfn 6555 ‘cfv 6560 ωcom 7888 1oc1o 8500 t++cttrcl 9748 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-opab 5205 df-xp 5690 df-rel 5691 df-ttrcl 9749 | 
| This theorem is referenced by: brttrcl 9754 ttrclss 9761 ttrclexg 9764 | 
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