Theorem ttrcleq 9676

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 5110	. . . . . . 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 1469	. . . . 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 1950	. . . 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 5176	. 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 9675	. 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 9675	. 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 2822	1 ⊢ (𝑅 = 𝑆 → t++𝑅 = t++𝑆)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1102   = wceq 1569  ∃wex 1808  ∀wral 3078  ∃wrex 3088   ∖ cdif 3901  ∅c0 4285   class class class wbr 5108  {copab 5172  suc csuc 6362   Fn wfn 6531  ‘cfv 6536  ωcom 7860  1_oc1o 8444  t++cttrcl 9674

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-br 5109  df-opab 5173  df-ttrcl 9675

This theorem is used by:  fineqvnttrclse  35545