Theorem ttrclresv 9646

Description: The transitive closure of 𝑅 restricted to V is the same as the transitive closure of 𝑅 itself. (Contributed by Scott Fenton, 20-Oct-2024.)

Assertion

Ref	Expression
ttrclresv	⊢ t++(𝑅 ↾ V) = t++𝑅

Proof of Theorem ttrclresv

Dummy variables 𝑓 𝑛 𝑎 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6853	. . . . . . . 8 ⊢ (𝑓‘𝑎) ∈ V
2		fvex 6853	. . . . . . . . 9 ⊢ (𝑓‘suc 𝑎) ∈ V
3	2	brresi 5948	. . . . . . . 8 ⊢ ((𝑓‘𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎) ↔ ((𝑓‘𝑎) ∈ V ∧ (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))
4	1, 3	mpbiran 709	. . . . . . 7 ⊢ ((𝑓‘𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎) ↔ (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))
5	4	ralbii 3075	. . . . . 6 ⊢ (∀𝑎 ∈ 𝑛 (𝑓‘𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎) ↔ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))
6	5	3anbi3i 1159	. . . . 5 ⊢ ((𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎)) ↔ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))
7	6	exbii 1848	. . . 4 ⊢ (∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎)) ↔ ∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))
8	7	rexbii 3076	. . 3 ⊢ (∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎)) ↔ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))
9	8	opabbii 5169	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))}
10		df-ttrcl 9637	. 2 ⊢ t++(𝑅 ↾ V) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎))}
11		df-ttrcl 9637	. 2 ⊢ t++𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))}
12	9, 10, 11	3eqtr4i 2762	1 ⊢ t++(𝑅 ↾ V) = t++𝑅

Syntax hints:   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3444   ∖ cdif 3908  ∅c0 4292   class class class wbr 5102  {copab 5164   ↾ cres 5633  suc csuc 6322   Fn wfn 6494  ‘cfv 6499  ωcom 7822  1_oc1o 8404  t++cttrcl 9636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-xp 5637  df-res 5643  df-iota 6452  df-fv 6507  df-ttrcl 9637

This theorem is referenced by:  ttrclco  9647  cottrcl  9648  dmttrcl  9650  rnttrcl  9651