Theorem brttrcl2 9626

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 9625	. 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 8395	. . . . . . . . 9 ⊢ 1o = suc ∅
3	2	difeq2i 4054	. . . . . . . 8 ⊢ (ω ∖ 1o) = (ω ∖ suc ∅)
4	3	eleq2i 2831	. . . . . . 7 ⊢ (𝑚 ∈ (ω ∖ 1o) ↔ 𝑚 ∈ (ω ∖ suc ∅))
5		peano1 7829	. . . . . . . 8 ⊢ ∅ ∈ ω
6		eldifsucnn 8590	. . . . . . . 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 4306	. . . . . . . 8 ⊢ (ω ∖ ∅) = ω
9	8	rexeqi 3296	. . . . . . 7 ⊢ (∃𝑛 ∈ (ω ∖ ∅)𝑚 = suc 𝑛 ↔ ∃𝑛 ∈ ω 𝑚 = suc 𝑛)
10	4, 7, 9	3bitri 298	. . . . . 6 ⊢ (𝑚 ∈ (ω ∖ 1o) ↔ ∃𝑛 ∈ ω 𝑚 = suc 𝑛)
11	10	anbi1i 630	. . . . 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 3169	. . . . 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 279	. . . 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 1855	. . 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 3064	. . 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 3266	. . 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 304	. 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 7749	. . . 4 ⊢ suc 𝑛 ∈ V
20		suceq 6378	. . . . . . 7 ⊢ (𝑚 = suc 𝑛 → suc 𝑚 = suc suc 𝑛)
21	20	fneq2d 6579	. . . . . 6 ⊢ (𝑚 = suc 𝑛 → (𝑓 Fn suc 𝑚 ↔ 𝑓 Fn suc suc 𝑛))
22		fveqeq2 6836	. . . . . . 7 ⊢ (𝑚 = suc 𝑛 → ((𝑓‘𝑚) = 𝐵 ↔ (𝑓‘suc 𝑛) = 𝐵))
23	22	anbi2d 636	. . . . . 6 ⊢ (𝑚 = suc 𝑛 → (((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑚) = 𝐵) ↔ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘suc 𝑛) = 𝐵)))
24		raleq 3294	. . . . . 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 1444	. . . . 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 3479	. . 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 3086	. 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 298	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 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063   ∖ cdif 3880  ∅c0 4261   class class class wbr 5072  suc csuc 6312   Fn wfn 6480  ‘cfv 6485  ωcom 7806  1_oc1o 8388  t++cttrcl 9619

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-ttrcl 9620

This theorem is referenced by:  ttrclss  9632  ttrclse  9639  fineqvnttrclse  35305