Theorem brttrcl2 9442

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 9441	. 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 8282	. . . . . . . . 9 ⊢ 1o = suc ∅
3	2	difeq2i 4059	. . . . . . . 8 ⊢ (ω ∖ 1o) = (ω ∖ suc ∅)
4	3	eleq2i 2832	. . . . . . 7 ⊢ (𝑚 ∈ (ω ∖ 1o) ↔ 𝑚 ∈ (ω ∖ suc ∅))
5		peano1 7724	. . . . . . . 8 ⊢ ∅ ∈ ω
6		eldifsucnn 8469	. . . . . . . 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 4312	. . . . . . . 8 ⊢ (ω ∖ ∅) = ω
9	8	rexeqi 3346	. . . . . . 7 ⊢ (∃𝑛 ∈ (ω ∖ ∅)𝑚 = suc 𝑛 ↔ ∃𝑛 ∈ ω 𝑚 = suc 𝑛)
10	4, 7, 9	3bitri 297	. . . . . 6 ⊢ (𝑚 ∈ (ω ∖ 1o) ↔ ∃𝑛 ∈ ω 𝑚 = suc 𝑛)
11	10	anbi1i 624	. . . . 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 3276	. . . . 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 277	. . . 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 1854	. . 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 3072	. . 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 3182	. . 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 303	. 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 3435	. . . . 5 ⊢ 𝑛 ∈ V
19	18	sucex 7647	. . . 4 ⊢ suc 𝑛 ∈ V
20		suceq 6329	. . . . . . 7 ⊢ (𝑚 = suc 𝑛 → suc 𝑚 = suc suc 𝑛)
21	20	fneq2d 6524	. . . . . 6 ⊢ (𝑚 = suc 𝑛 → (𝑓 Fn suc 𝑚 ↔ 𝑓 Fn suc suc 𝑛))
22		fveqeq2 6778	. . . . . . 7 ⊢ (𝑚 = suc 𝑛 → ((𝑓‘𝑚) = 𝐵 ↔ (𝑓‘suc 𝑛) = 𝐵))
23	22	anbi2d 629	. . . . . 6 ⊢ (𝑚 = suc 𝑛 → (((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑚) = 𝐵) ↔ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘suc 𝑛) = 𝐵)))
24		raleq 3341	. . . . . 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 1435	. . . . 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 1928	. . . 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 3478	. . 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 3180	. 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 297	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 205   ∧ wa 396   ∧ w3a 1086   = wceq 1542  ∃wex 1786   ∈ wcel 2110  ∀wral 3066  ∃wrex 3067   ∖ cdif 3889  ∅c0 4262   class class class wbr 5079  suc csuc 6266   Fn wfn 6426  ‘cfv 6431  ωcom 7701  1_oc1o 8275  t++cttrcl 9435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7580

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6200  df-ord 6267  df-on 6268  df-lim 6269  df-suc 6270  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-ov 7272  df-oprab 7273  df-mpo 7274  df-om 7702  df-2nd 7819  df-frecs 8082  df-wrecs 8113  df-recs 8187  df-rdg 8226  df-1o 8282  df-oadd 8286  df-ttrcl 9436

This theorem is referenced by:  ttrclss  9448  ttrclse  9455