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Theorem ttrcleq 9747
Description: Equality theorem for transitive closure. (Contributed by Scott Fenton, 17-Oct-2024.)
Assertion
Ref Expression
ttrcleq (𝑅 = 𝑆 → t++𝑅 = t++𝑆)

Proof of Theorem ttrcleq
Dummy variables 𝑓 𝑚 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq 5150 . . . . . . 7 (𝑅 = 𝑆 → ((𝑓𝑚)𝑅(𝑓‘suc 𝑚) ↔ (𝑓𝑚)𝑆(𝑓‘suc 𝑚)))
21ralbidv 3176 . . . . . 6 (𝑅 = 𝑆 → (∀𝑚𝑛 (𝑓𝑚)𝑅(𝑓‘suc 𝑚) ↔ ∀𝑚𝑛 (𝑓𝑚)𝑆(𝑓‘suc 𝑚)))
323anbi3d 1441 . . . . 5 (𝑅 = 𝑆 → ((𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑚𝑛 (𝑓𝑚)𝑅(𝑓‘suc 𝑚)) ↔ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑚𝑛 (𝑓𝑚)𝑆(𝑓‘suc 𝑚))))
43exbidv 1919 . . . 4 (𝑅 = 𝑆 → (∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑚𝑛 (𝑓𝑚)𝑅(𝑓‘suc 𝑚)) ↔ ∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑚𝑛 (𝑓𝑚)𝑆(𝑓‘suc 𝑚))))
54rexbidv 3177 . . 3 (𝑅 = 𝑆 → (∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑚𝑛 (𝑓𝑚)𝑅(𝑓‘suc 𝑚)) ↔ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑚𝑛 (𝑓𝑚)𝑆(𝑓‘suc 𝑚))))
65opabbidv 5214 . 2 (𝑅 = 𝑆 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑚𝑛 (𝑓𝑚)𝑅(𝑓‘suc 𝑚))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑚𝑛 (𝑓𝑚)𝑆(𝑓‘suc 𝑚))})
7 df-ttrcl 9746 . 2 t++𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑚𝑛 (𝑓𝑚)𝑅(𝑓‘suc 𝑚))}
8 df-ttrcl 9746 . 2 t++𝑆 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑚𝑛 (𝑓𝑚)𝑆(𝑓‘suc 𝑚))}
96, 7, 83eqtr4g 2800 1 (𝑅 = 𝑆 → t++𝑅 = t++𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wex 1776  wral 3059  wrex 3068  cdif 3960  c0 4339   class class class wbr 5148  {copab 5210  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-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-br 5149  df-opab 5211  df-ttrcl 9746
This theorem is referenced by: (None)
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