Theorem brttrcl 9680

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 9679	. . 3 ⊢ Rel t++𝑅
2	1	brrelex12i 5715	. 2 ⊢ (𝐴t++𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		fvex 6894	. . . . . . 7 ⊢ (𝑓‘∅) ∈ V
4		eleq1 2850	. . . . . . 7 ⊢ ((𝑓‘∅) = 𝐴 → ((𝑓‘∅) ∈ V ↔ 𝐴 ∈ V))
5	3, 4	mpbii 236	. . . . . 6 ⊢ ((𝑓‘∅) = 𝐴 → 𝐴 ∈ V)
6		fvex 6894	. . . . . . 7 ⊢ (𝑓‘𝑛) ∈ V
7		eleq1 2850	. . . . . . 7 ⊢ ((𝑓‘𝑛) = 𝐵 → ((𝑓‘𝑛) ∈ V ↔ 𝐵 ∈ V))
8	6, 7	mpbii 236	. . . . . 6 ⊢ ((𝑓‘𝑛) = 𝐵 → 𝐵 ∈ V)
9	5, 8	anim12i 624	. . . . 5 ⊢ (((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
10	9	3ad2ant2 1151	. . . 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 1959	. . 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 3161	. 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 2774	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑓‘∅) = 𝑥 ↔ (𝑓‘∅) = 𝐴))
14	13	anbi1d 642	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ↔ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝑦)))
15	14	3anbi2d 1468	. . . . 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 1950	. . . 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 3188	. . 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 2774	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝑓‘𝑛) = 𝑦 ↔ (𝑓‘𝑛) = 𝐵))
19	18	anbi2d 641	. . . . . 6 ⊢ (𝑦 = 𝐵 → (((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝑦) ↔ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝐵)))
20	19	3anbi2d 1468	. . . . 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 1950	. . . 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 3188	. . 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 9675	. . 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 5523	. 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 381	1 ⊢ (𝐴t++𝑅𝐵 ↔ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝐵) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569  ∃wex 1808   ∈ wcel 2142  ∀wral 3078  ∃wrex 3088  Vcvv 3454   ∖ cdif 3901  ∅c0 4285   class class class wbr 5108  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-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-xp 5666  df-rel 5667  df-iota 6492  df-fv 6544  df-ttrcl 9675

This theorem is used by:  brttrcl2  9681  ssttrcl  9682  ttrcltr  9683