Theorem ttrcleq 33695

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.

Step	Hyp	Ref	Expression
1		breq 5072	. . . . . . 7 ⊢ (𝑅 = 𝑆 → ((𝑓‘𝑚)𝑅(𝑓‘suc 𝑚) ↔ (𝑓‘𝑚)𝑆(𝑓‘suc 𝑚)))
2	1	ralbidv 3120	. . . . . 6 ⊢ (𝑅 = 𝑆 → (∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑅(𝑓‘suc 𝑚) ↔ ∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑆(𝑓‘suc 𝑚)))
3	2	3anbi3d 1440	. . . . 5 ⊢ (𝑅 = 𝑆 → ((𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑅(𝑓‘suc 𝑚)) ↔ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑆(𝑓‘suc 𝑚))))
4	3	exbidv 1925	. . . 4 ⊢ (𝑅 = 𝑆 → (∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑅(𝑓‘suc 𝑚)) ↔ ∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑆(𝑓‘suc 𝑚))))
5	4	rexbidv 3225	. . 3 ⊢ (𝑅 = 𝑆 → (∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑅(𝑓‘suc 𝑚)) ↔ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑆(𝑓‘suc 𝑚))))
6	5	opabbidv 5136	. 2 ⊢ (𝑅 = 𝑆 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑅(𝑓‘suc 𝑚))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑆(𝑓‘suc 𝑚))})
7		df-ttrcl 33694	. 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))}
8		df-ttrcl 33694	. 2 ⊢ t++𝑆 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑆(𝑓‘suc 𝑚))}
9	6, 7, 8	3eqtr4g 2804	1 ⊢ (𝑅 = 𝑆 → t++𝑅 = t++𝑆)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539  ∃wex 1783  ∀wral 3063  ∃wrex 3064   ∖ cdif 3880  ∅c0 4253   class class class wbr 5070  {copab 5132  suc csuc 6253   Fn wfn 6413  ‘cfv 6418  ωcom 7687  1_oc1o 8260  t++cttrcl 33693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-br 5071  df-opab 5133  df-ttrcl 33694

This theorem is referenced by: (None)