Theorem brttrcl2 9650

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 9649	. 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 8412	. . . . . . . . 9 ⊢ 1o = suc ∅
3	2	difeq2i 4079	. . . . . . . 8 ⊢ (ω ∖ 1o) = (ω ∖ suc ∅)
4	3	eleq2i 2829	. . . . . . 7 ⊢ (𝑚 ∈ (ω ∖ 1o) ↔ 𝑚 ∈ (ω ∖ suc ∅))
5		peano1 7825	. . . . . . . 8 ⊢ ∅ ∈ ω
6		eldifsucnn 8610	. . . . . . . 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 4332	. . . . . . . 8 ⊢ (ω ∖ ∅) = ω
9	8	rexeqi 3312	. . . . . . 7 ⊢ (∃𝑛 ∈ (ω ∖ ∅)𝑚 = suc 𝑛 ↔ ∃𝑛 ∈ ω 𝑚 = suc 𝑛)
10	4, 7, 9	3bitri 296	. . . . . 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 3185	. . . . 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 1850	. . 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 3074	. . 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 3271	. . 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 302	. 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 3449	. . . . 5 ⊢ 𝑛 ∈ V
19	18	sucex 7741	. . . 4 ⊢ suc 𝑛 ∈ V
20		suceq 6383	. . . . . . 7 ⊢ (𝑚 = suc 𝑛 → suc 𝑚 = suc suc 𝑛)
21	20	fneq2d 6596	. . . . . 6 ⊢ (𝑚 = suc 𝑛 → (𝑓 Fn suc 𝑚 ↔ 𝑓 Fn suc suc 𝑛))
22		fveqeq2 6851	. . . . . . 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 3309	. . . . . 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 1436	. . . . 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 1924	. . . 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 3494	. . 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 3097	. 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 296	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 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073   ∖ cdif 3907  ∅c0 4282   class class class wbr 5105  suc csuc 6319   Fn wfn 6491  ‘cfv 6496  ωcom 7802  1_oc1o 8405  t++cttrcl 9643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-ttrcl 9644

This theorem is referenced by:  ttrclss  9656  ttrclse  9663