Theorem brttrcl2 9683

Description: Characterization of elements of the transitive closure of a relation. (Contributed by Scott Fenton, 24-Aug-2024.)

Assertion

Ref	Expression
brttrcl2	⊢ (𝐴t++𝑅𝐵 ↔ ∃𝑛 ∈ ω ∃𝑓(𝑓 Fn suc suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘suc 𝑛) = 𝐵) ∧ ∀𝑎 ∈ suc 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 brttrcl2

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brttrcl 9682	. 2 ⊢ (𝐴t++𝑅𝐵 ↔ ∃𝑚 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑚) = 𝐵) ∧ ∀𝑎 ∈ 𝑚 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))
2		df-1o 8453	. . . . . . . . 9 ⊢ 1o = suc ∅
3	2	difeq2i 4086	. . . . . . . 8 ⊢ (ω ∖ 1o) = (ω ∖ suc ∅)
4	3	eleq2i 2861	. . . . . . 7 ⊢ (𝑚 ∈ (ω ∖ 1o) ↔ 𝑚 ∈ (ω ∖ suc ∅))
5		peano1 7885	. . . . . . . 8 ⊢ ∅ ∈ ω
6		eldifsucnn 8650	. . . . . . . 8 ⊢ (∅ ∈ ω → (𝑚 ∈ (ω ∖ suc ∅) ↔ ∃𝑛 ∈ (ω ∖ ∅)𝑚 = suc 𝑛))
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ (𝑚 ∈ (ω ∖ suc ∅) ↔ ∃𝑛 ∈ (ω ∖ ∅)𝑚 = suc 𝑛)
8		dif0 4341	. . . . . . . 8 ⊢ (ω ∖ ∅) = ω
9	8	rexeqi 3328	. . . . . . 7 ⊢ (∃𝑛 ∈ (ω ∖ ∅)𝑚 = suc 𝑛 ↔ ∃𝑛 ∈ ω 𝑚 = suc 𝑛)
10	4, 7, 9	3bitri 300	. . . . . 6 ⊢ (𝑚 ∈ (ω ∖ 1o) ↔ ∃𝑛 ∈ ω 𝑚 = suc 𝑛)
11	10	anbi1i 635	. . . . 5 ⊢ ((𝑚 ∈ (ω ∖ 1o) ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑚) = 𝐵) ∧ ∀𝑎 ∈ 𝑚 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) ↔ (∃𝑛 ∈ ω 𝑚 = suc 𝑛 ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑚) = 𝐵) ∧ ∀𝑎 ∈ 𝑚 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))))
12		r19.41v 3201	. . . . 5 ⊢ (∃𝑛 ∈ ω (𝑚 = suc 𝑛 ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑚) = 𝐵) ∧ ∀𝑎 ∈ 𝑚 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) ↔ (∃𝑛 ∈ ω 𝑚 = suc 𝑛 ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑚) = 𝐵) ∧ ∀𝑎 ∈ 𝑚 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))))
13	11, 12	bitr4i 281	. . . 4 ⊢ ((𝑚 ∈ (ω ∖ 1o) ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑚) = 𝐵) ∧ ∀𝑎 ∈ 𝑚 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) ↔ ∃𝑛 ∈ ω (𝑚 = suc 𝑛 ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑚) = 𝐵) ∧ ∀𝑎 ∈ 𝑚 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))))
14	13	exbii 1875	. . 3 ⊢ (∃𝑚(𝑚 ∈ (ω ∖ 1o) ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑚) = 𝐵) ∧ ∀𝑎 ∈ 𝑚 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) ↔ ∃𝑚∃𝑛 ∈ ω (𝑚 = suc 𝑛 ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑚) = 𝐵) ∧ ∀𝑎 ∈ 𝑚 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))))
15		df-rex 3096	. . 3 ⊢ (∃𝑚 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑚) = 𝐵) ∧ ∀𝑎 ∈ 𝑚 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) ↔ ∃𝑚(𝑚 ∈ (ω ∖ 1o) ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑚) = 𝐵) ∧ ∀𝑎 ∈ 𝑚 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))))
16		rexcom4 3298	. . 3 ⊢ (∃𝑛 ∈ ω ∃𝑚(𝑚 = suc 𝑛 ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑚) = 𝐵) ∧ ∀𝑎 ∈ 𝑚 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) ↔ ∃𝑚∃𝑛 ∈ ω (𝑚 = suc 𝑛 ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑚) = 𝐵) ∧ ∀𝑎 ∈ 𝑚 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))))
17	14, 15, 16	3bitr4i 306	. 2 ⊢ (∃𝑚 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑚) = 𝐵) ∧ ∀𝑎 ∈ 𝑚 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) ↔ ∃𝑛 ∈ ω ∃𝑚(𝑚 = suc 𝑛 ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑚) = 𝐵) ∧ ∀𝑎 ∈ 𝑚 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))))
18		vex 3467	. . . . 5 ⊢ 𝑛 ∈ V
19	18	sucex 7805	. . . 4 ⊢ suc 𝑛 ∈ V
20		suceq 6430	. . . . . . 7 ⊢ (𝑚 = suc 𝑛 → suc 𝑚 = suc suc 𝑛)
21	20	fneq2d 6630	. . . . . 6 ⊢ (𝑚 = suc 𝑛 → (𝑓 Fn suc 𝑚 ↔ 𝑓 Fn suc suc 𝑛))
22		fveqeq2 6891	. . . . . . 7 ⊢ (𝑚 = suc 𝑛 → ((𝑓‘𝑚) = 𝐵 ↔ (𝑓‘suc 𝑛) = 𝐵))
23	22	anbi2d 641	. . . . . 6 ⊢ (𝑚 = suc 𝑛 → (((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑚) = 𝐵) ↔ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘suc 𝑛) = 𝐵)))
24		raleq 3326	. . . . . 6 ⊢ (𝑚 = suc 𝑛 → (∀𝑎 ∈ 𝑚 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎) ↔ ∀𝑎 ∈ suc 𝑛(𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))
25	21, 23, 24	3anbi123d 1462	. . . . 5 ⊢ (𝑚 = suc 𝑛 → ((𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑚) = 𝐵) ∧ ∀𝑎 ∈ 𝑚 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) ↔ (𝑓 Fn suc suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘suc 𝑛) = 𝐵) ∧ ∀𝑎 ∈ suc 𝑛(𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))))
26	25	exbidv 1948	. . . 4 ⊢ (𝑚 = suc 𝑛 → (∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑚) = 𝐵) ∧ ∀𝑎 ∈ 𝑚 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) ↔ ∃𝑓(𝑓 Fn suc suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘suc 𝑛) = 𝐵) ∧ ∀𝑎 ∈ suc 𝑛(𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))))
27	19, 26	ceqsexv 3511	. . 3 ⊢ (∃𝑚(𝑚 = suc 𝑛 ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑚) = 𝐵) ∧ ∀𝑎 ∈ 𝑚 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) ↔ ∃𝑓(𝑓 Fn suc suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘suc 𝑛) = 𝐵) ∧ ∀𝑎 ∈ suc 𝑛(𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))
28	27	rexbii 3118	. 2 ⊢ (∃𝑛 ∈ ω ∃𝑚(𝑚 = suc 𝑛 ∧ ∃𝑓(𝑓 Fn suc 𝑚 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑚) = 𝐵) ∧ ∀𝑎 ∈ 𝑚 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) ↔ ∃𝑛 ∈ ω ∃𝑓(𝑓 Fn suc suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘suc 𝑛) = 𝐵) ∧ ∀𝑎 ∈ suc 𝑛(𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))
29	1, 17, 28	3bitri 300	1 ⊢ (𝐴t++𝑅𝐵 ↔ ∃𝑛 ∈ ω ∃𝑓(𝑓 Fn suc suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘suc 𝑛) = 𝐵) ∧ ∀𝑎 ∈ suc 𝑛(𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095   ∖ cdif 3910  ∅c0 4294   class class class wbr 5113  suc csuc 6363   Fn wfn 6532  ‘cfv 6537  ωcom 7862  1_oc1o 8446  t++cttrcl 9676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-oadd 8457  df-ttrcl 9677

This theorem is referenced by:  ttrclss  9689  ttrclse  9696  fineqvnttrclse  35460