Theorem dfrtrclrec2 15020

Description: If two elements are connected by a reflexive, transitive closure, then they are connected via 𝑛 instances the relation, for some 𝑛. (Contributed by Drahflow, 12-Nov-2015.) (Revised by AV, 13-Jul-2024.)

Hypothesis

Ref	Expression
dfrtrclrec2.1	⊢ (𝜑 → Rel 𝑅)

Assertion

Ref	Expression
dfrtrclrec2	⊢ (𝜑 → (𝐴(t*rec‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵))

Distinct variable groups:   𝑅,𝑛   𝐴,𝑛   𝐵,𝑛

Allowed substitution hint:   𝜑(𝑛)

Proof of Theorem dfrtrclrec2

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝑅 ∈ V)
2		nn0ex 12443	. . . . . . 7 ⊢ ℕ0 ∈ V
3		ovex 7400	. . . . . . 7 ⊢ (𝑅↑𝑟𝑛) ∈ V
4	2, 3	iunex 7921	. . . . . 6 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∈ V
5		oveq1 7374	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛))
6	5	iuneq2d 4964	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
7		eqid 2736	. . . . . . 7 ⊢ (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
8	6, 7	fvmptg 6945	. . . . . 6 ⊢ ((𝑅 ∈ V ∧ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∈ V) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
9	1, 4, 8	sylancl 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
10	9	ex 412	. . . 4 ⊢ (𝜑 → (𝑅 ∈ V → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)))
11		iun0 5004	. . . . . 6 ⊢ ∪ 𝑛 ∈ ℕ0 ∅ = ∅
12	11	a1i 11	. . . . 5 ⊢ (¬ 𝑅 ∈ V → ∪ 𝑛 ∈ ℕ0 ∅ = ∅)
13		reldmrelexp 14983	. . . . . . 7 ⊢ Rel dom ↑𝑟
14	13	ovprc1 7406	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → (𝑅↑𝑟𝑛) = ∅)
15	14	iuneq2d 4964	. . . . 5 ⊢ (¬ 𝑅 ∈ V → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 ∅)
16		fvprc 6832	. . . . 5 ⊢ (¬ 𝑅 ∈ V → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∅)
17	12, 15, 16	3eqtr4rd 2782	. . . 4 ⊢ (¬ 𝑅 ∈ V → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
18	10, 17	pm2.61d1 180	. . 3 ⊢ (𝜑 → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
19		breq 5087	. . . 4 ⊢ (((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) → (𝐴((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)𝐵 ↔ 𝐴∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)𝐵))
20		eliun 4937	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ↔ ∃𝑛 ∈ ℕ0 ⟨𝐴, 𝐵⟩ ∈ (𝑅↑𝑟𝑛))
21	20	a1i 11	. . . . 5 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ↔ ∃𝑛 ∈ ℕ0 ⟨𝐴, 𝐵⟩ ∈ (𝑅↑𝑟𝑛)))
22		df-br 5086	. . . . 5 ⊢ (𝐴∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
23		df-br 5086	. . . . . 6 ⊢ (𝐴(𝑅↑𝑟𝑛)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅↑𝑟𝑛))
24	23	rexbii 3084	. . . . 5 ⊢ (∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵 ↔ ∃𝑛 ∈ ℕ0 ⟨𝐴, 𝐵⟩ ∈ (𝑅↑𝑟𝑛))
25	21, 22, 24	3bitr4g 314	. . . 4 ⊢ (𝜑 → (𝐴∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵))
26	19, 25	sylan9bb 509	. . 3 ⊢ ((((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∧ 𝜑) → (𝐴((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵))
27	18, 26	mpancom 689	. 2 ⊢ (𝜑 → (𝐴((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵))
28		df-rtrclrec 15018	. . 3 ⊢ t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
29		fveq1 6839	. . . . . 6 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅))
30	29	breqd 5096	. . . . 5 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → (𝐴(t*rec‘𝑅)𝐵 ↔ 𝐴((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)𝐵))
31	30	bibi1d 343	. . . 4 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → ((𝐴(t*rec‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵) ↔ (𝐴((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵)))
32	31	imbi2d 340	. . 3 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → ((𝜑 → (𝐴(t*rec‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵)) ↔ (𝜑 → (𝐴((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵))))
33	28, 32	ax-mp 5	. 2 ⊢ ((𝜑 → (𝐴(t*rec‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵)) ↔ (𝜑 → (𝐴((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵)))
34	27, 33	mpbir 231	1 ⊢ (𝜑 → (𝐴(t*rec‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3061  Vcvv 3429  ∅c0 4273  ⟨cop 4573  ∪ ciun 4933   class class class wbr 5085   ↦ cmpt 5166  Rel wrel 5636  ‘cfv 6498  (class class class)co 7367  ℕ₀cn0 12437  ↑_𝑟crelexp 14981  t*reccrtrcl 15017

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-1cn 11096  ax-addcl 11098

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-nn 12175  df-n0 12438  df-relexp 14982  df-rtrclrec 15018

This theorem is referenced by:  rtrclreclem3  15022  rtrclind  15027