Theorem ttrcleq 9666

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 5104	. . . . . . 7 ⊢ (𝑅 = 𝑆 → ((𝑓‘𝑚)𝑅(𝑓‘suc 𝑚) ↔ (𝑓‘𝑚)𝑆(𝑓‘suc 𝑚)))
2	1	ralbidv 3187	. . . . . 6 ⊢ (𝑅 = 𝑆 → (∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑅(𝑓‘suc 𝑚) ↔ ∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑆(𝑓‘suc 𝑚)))
3	2	3anbi3d 1465	. . . . 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 1943	. . . 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 3188	. . 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 5168	. 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 9665	. 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 9665	. 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 2824	1 ⊢ (𝑅 = 𝑆 → t++𝑅 = t++𝑆)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1562  ∃wex 1801  ∀wral 3078  ∃wrex 3088   ∖ cdif 3903  ∅c0 4287   class class class wbr 5102  {copab 5164  suc csuc 6350   Fn wfn 6518  ‘cfv 6523  ωcom 7848  1_oc1o 8432  t++cttrcl 9664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1101  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-br 5103  df-opab 5165  df-ttrcl 9665

This theorem is referenced by:  fineqvnttrclse  35424