Theorem brttrcl 9708

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 9707	. . 3 ⊢ Rel t++𝑅
2	1	brrelex12i 5732	. 2 ⊢ (𝐴t++𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		fvex 6905	. . . . . . 7 ⊢ (𝑓‘∅) ∈ V
4		eleq1 2822	. . . . . . 7 ⊢ ((𝑓‘∅) = 𝐴 → ((𝑓‘∅) ∈ V ↔ 𝐴 ∈ V))
5	3, 4	mpbii 232	. . . . . 6 ⊢ ((𝑓‘∅) = 𝐴 → 𝐴 ∈ V)
6		fvex 6905	. . . . . . 7 ⊢ (𝑓‘𝑛) ∈ V
7		eleq1 2822	. . . . . . 7 ⊢ ((𝑓‘𝑛) = 𝐵 → ((𝑓‘𝑛) ∈ V ↔ 𝐵 ∈ V))
8	6, 7	mpbii 232	. . . . . 6 ⊢ ((𝑓‘𝑛) = 𝐵 → 𝐵 ∈ V)
9	5, 8	anim12i 614	. . . . 5 ⊢ (((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
10	9	3ad2ant2 1135	. . . 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 1934	. . 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 3152	. 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 2745	. . . . . . 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 1442	. . . . 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 1925	. . . 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 3179	. . 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 2745	. . . . . . 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 1442	. . . . 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 1925	. . . 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 3179	. . 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 9703	. . 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 5540	. 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 380	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 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  ∃wex 1782   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071  Vcvv 3475   ∖ cdif 3946  ∅c0 4323   class class class wbr 5149  suc csuc 6367   Fn wfn 6539  ‘cfv 6544  ωcom 7855  1_oc1o 8459  t++cttrcl 9702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-iota 6496  df-fv 6552  df-ttrcl 9703

This theorem is referenced by:  brttrcl2  9709  ssttrcl  9710  ttrcltr  9711