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Theorem ttrcleq 9704
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 5151 . . . . . . 7 (𝑅 = 𝑆 → ((𝑓𝑚)𝑅(𝑓‘suc 𝑚) ↔ (𝑓𝑚)𝑆(𝑓‘suc 𝑚)))
21ralbidv 3178 . . . . . 6 (𝑅 = 𝑆 → (∀𝑚𝑛 (𝑓𝑚)𝑅(𝑓‘suc 𝑚) ↔ ∀𝑚𝑛 (𝑓𝑚)𝑆(𝑓‘suc 𝑚)))
323anbi3d 1443 . . . . 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 1925 . . . 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 3179 . . 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 5215 . 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 9703 . 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 9703 . 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 2798 1 (𝑅 = 𝑆 → t++𝑅 = t++𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wex 1782  wral 3062  wrex 3071  cdif 3946  c0 4323   class class class wbr 5149  {copab 5211  suc csuc 6367   Fn wfn 6539  cfv 6544  ωcom 7855  1oc1o 8459  t++cttrcl 9702
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-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-br 5150  df-opab 5212  df-ttrcl 9703
This theorem is referenced by: (None)
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