Theorem brttrcl 9695

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 9694	. . 3 ⊢ Rel t++𝑅
2	1	brrelex12i 5714	. 2 ⊢ (𝐴t++𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		fvex 6895	. . . . . . 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 6895	. . . . . . 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 625	. . . . 5 ⊢ (((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
10	9	3ad2ant2 1152	. . . 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 1963	. . 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 643	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ↔ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝑦)))
15	14	3anbi2d 1469	. . . . 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 1954	. . . 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 642	. . . . . 6 ⊢ (𝑦 = 𝐵 → (((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝑦) ↔ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝐵)))
20	19	3anbi2d 1469	. . . . 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 1954	. . . 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 9690	. . 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 5522	. 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 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088  Vcvv 3453   ∖ cdif 3899  ∅c0 4282   class class class wbr 5107  suc csuc 6363   Fn wfn 6532  ‘cfv 6537  ωcom 7865  1_oc1o 8451  t++cttrcl 9689

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-iota 6493  df-fv 6545  df-ttrcl 9690

This theorem is used by:  brttrcl2  9696  ssttrcl  9697  ttrcltr  9698