Theorem ttrcleq 9467

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 5076	. . . . . . 7 ⊢ (𝑅 = 𝑆 → ((𝑓‘𝑚)𝑅(𝑓‘suc 𝑚) ↔ (𝑓‘𝑚)𝑆(𝑓‘suc 𝑚)))
2	1	ralbidv 3112	. . . . . 6 ⊢ (𝑅 = 𝑆 → (∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑅(𝑓‘suc 𝑚) ↔ ∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑆(𝑓‘suc 𝑚)))
3	2	3anbi3d 1441	. . . . 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 1924	. . . 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 3226	. . 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 5140	. 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 9466	. 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 9466	. 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 2803	1 ⊢ (𝑅 = 𝑆 → t++𝑅 = t++𝑆)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782  ∀wral 3064  ∃wrex 3065   ∖ cdif 3884  ∅c0 4256   class class class wbr 5074  {copab 5136  suc csuc 6268   Fn wfn 6428  ‘cfv 6433  ωcom 7712  1_oc1o 8290  t++cttrcl 9465

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-br 5075  df-opab 5137  df-ttrcl 9466

This theorem is referenced by: (None)