Theorem brttrcl 9732

Description: Characterization of elements of the transitive closure of a relation. (Contributed by Scott Fenton, 18-Aug-2024.)

Assertion

Ref	Expression
brttrcl	⊢ (𝐴t++𝑅𝐵 ↔ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝐵) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))

Distinct variable groups:   𝐴,𝑛,𝑓,𝑎   𝐵,𝑛,𝑓,𝑎   𝑅,𝑛,𝑓,𝑎

Proof of Theorem brttrcl

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

Step	Hyp	Ref	Expression
1		relttrcl 9731	. . 3 ⊢ Rel t++𝑅
2	1	brrelex12i 5714	. 2 ⊢ (𝐴t++𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		fvex 6894	. . . . . . 7 ⊢ (𝑓‘∅) ∈ V
4		eleq1 2823	. . . . . . 7 ⊢ ((𝑓‘∅) = 𝐴 → ((𝑓‘∅) ∈ V ↔ 𝐴 ∈ V))
5	3, 4	mpbii 233	. . . . . 6 ⊢ ((𝑓‘∅) = 𝐴 → 𝐴 ∈ V)
6		fvex 6894	. . . . . . 7 ⊢ (𝑓‘𝑛) ∈ V
7		eleq1 2823	. . . . . . 7 ⊢ ((𝑓‘𝑛) = 𝐵 → ((𝑓‘𝑛) ∈ V ↔ 𝐵 ∈ V))
8	6, 7	mpbii 233	. . . . . 6 ⊢ ((𝑓‘𝑛) = 𝐵 → 𝐵 ∈ V)
9	5, 8	anim12i 613	. . . . 5 ⊢ (((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
10	9	3ad2ant2 1134	. . . 4 ⊢ ((𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝐵) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
11	10	exlimiv 1930	. . 3 ⊢ (∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝐵) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
12	11	rexlimivw 3138	. 2 ⊢ (∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝐵) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
13		eqeq2 2748	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑓‘∅) = 𝑥 ↔ (𝑓‘∅) = 𝐴))
14	13	anbi1d 631	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ↔ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝑦)))
15	14	3anbi2d 1443	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) ↔ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))))
16	15	exbidv 1921	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) ↔ ∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))))
17	16	rexbidv 3165	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) ↔ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))))
18		eqeq2 2748	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝑓‘𝑛) = 𝑦 ↔ (𝑓‘𝑛) = 𝐵))
19	18	anbi2d 630	. . . . . 6 ⊢ (𝑦 = 𝐵 → (((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝑦) ↔ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝐵)))
20	19	3anbi2d 1443	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) ↔ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝐵) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))))
21	20	exbidv 1921	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) ↔ ∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝐵) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))))
22	21	rexbidv 3165	. . 3 ⊢ (𝑦 = 𝐵 → (∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) ↔ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝐵) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))))
23		df-ttrcl 9727	. . 3 ⊢ 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))}
24	17, 22, 23	brabg 5519	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴t++𝑅𝐵 ↔ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝐵) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))))
25	2, 12, 24	pm5.21nii 378	1 ⊢ (𝐴t++𝑅𝐵 ↔ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝐵) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061  Vcvv 3464   ∖ cdif 3928  ∅c0 4313   class class class wbr 5124  suc csuc 6359   Fn wfn 6531  ‘cfv 6536  ωcom 7866  1_oc1o 8478  t++cttrcl 9726

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-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

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 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-xp 5665  df-rel 5666  df-iota 6489  df-fv 6544  df-ttrcl 9727

This theorem is referenced by:  brttrcl2  9733  ssttrcl  9734  ttrcltr  9735