Theorem ttrclresv 9714

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 6898	. . . . . . . 8 ⊢ (𝑓‘𝑎) ∈ V
2		fvex 6898	. . . . . . . . 9 ⊢ (𝑓‘suc 𝑎) ∈ V
3	2	brresi 5984	. . . . . . . 8 ⊢ ((𝑓‘𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎) ↔ ((𝑓‘𝑎) ∈ V ∧ (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))
4	1, 3	mpbiran 706	. . . . . . 7 ⊢ ((𝑓‘𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎) ↔ (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))
5	4	ralbii 3087	. . . . . 6 ⊢ (∀𝑎 ∈ 𝑛 (𝑓‘𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎) ↔ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))
6	5	3anbi3i 1156	. . . . 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 1842	. . . 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 3088	. . 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 5208	. 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 9705	. 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 9705	. 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 2764	1 ⊢ t++(𝑅 ↾ V) = t++𝑅

Syntax hints:   ∧ wa 395   ∧ w3a 1084   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∀wral 3055  ∃wrex 3064  Vcvv 3468   ∖ cdif 3940  ∅c0 4317   class class class wbr 5141  {copab 5203   ↾ cres 5671  suc csuc 6360   Fn wfn 6532  ‘cfv 6537  ωcom 7852  1_oc1o 8460  t++cttrcl 9704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-res 5681  df-iota 6489  df-fv 6545  df-ttrcl 9705

This theorem is referenced by:  ttrclco  9715  cottrcl  9716  dmttrcl  9718  rnttrcl  9719