Theorem dfrtrclrec2 15031

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 12455	. . . . . . 7 ⊢ ℕ0 ∈ V
3		ovex 7423	. . . . . . 7 ⊢ (𝑅↑𝑟𝑛) ∈ V
4	2, 3	iunex 7950	. . . . . 6 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∈ V
5		oveq1 7397	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛))
6	5	iuneq2d 4989	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
7		eqid 2730	. . . . . . 7 ⊢ (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
8	6, 7	fvmptg 6969	. . . . . 6 ⊢ ((𝑅 ∈ V ∧ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∈ V) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
9	1, 4, 8	sylancl 586	. . . . 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 5029	. . . . . 6 ⊢ ∪ 𝑛 ∈ ℕ0 ∅ = ∅
12	11	a1i 11	. . . . 5 ⊢ (¬ 𝑅 ∈ V → ∪ 𝑛 ∈ ℕ0 ∅ = ∅)
13		reldmrelexp 14994	. . . . . . 7 ⊢ Rel dom ↑𝑟
14	13	ovprc1 7429	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → (𝑅↑𝑟𝑛) = ∅)
15	14	iuneq2d 4989	. . . . 5 ⊢ (¬ 𝑅 ∈ V → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 ∅)
16		fvprc 6853	. . . . 5 ⊢ (¬ 𝑅 ∈ V → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∅)
17	12, 15, 16	3eqtr4rd 2776	. . . 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 5112	. . . 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 4962	. . . . . 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 5111	. . . . 5 ⊢ (𝐴∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
23		df-br 5111	. . . . . 6 ⊢ (𝐴(𝑅↑𝑟𝑛)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅↑𝑟𝑛))
24	23	rexbii 3077	. . . . 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 688	. 2 ⊢ (𝜑 → (𝐴((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵))
28		df-rtrclrec 15029	. . 3 ⊢ t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
29		fveq1 6860	. . . . . 6 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅))
30	29	breqd 5121	. . . . 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 1540   ∈ wcel 2109  ∃wrex 3054  Vcvv 3450  ∅c0 4299  ⟨cop 4598  ∪ ciun 4958   class class class wbr 5110   ↦ cmpt 5191  Rel wrel 5646  ‘cfv 6514  (class class class)co 7390  ℕ₀cn0 12449  ↑_𝑟crelexp 14992  t*reccrtrcl 15028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-1cn 11133  ax-addcl 11135

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-nn 12194  df-n0 12450  df-relexp 14993  df-rtrclrec 15029

This theorem is referenced by:  rtrclreclem3  15033  rtrclind  15038