Theorem dfrtrclrec2 14955

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 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝑅 ∈ V)
2		nn0ex 12428	. . . . . . 7 ⊢ ℕ0 ∈ V
3		ovex 7395	. . . . . . 7 ⊢ (𝑅↑𝑟𝑛) ∈ V
4	2, 3	iunex 7906	. . . . . 6 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∈ V
5		oveq1 7369	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛))
6	5	iuneq2d 4988	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
7		eqid 2731	. . . . . . 7 ⊢ (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
8	6, 7	fvmptg 6951	. . . . . 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 413	. . . 4 ⊢ (𝜑 → (𝑅 ∈ V → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)))
11		iun0 5027	. . . . . 6 ⊢ ∪ 𝑛 ∈ ℕ0 ∅ = ∅
12	11	a1i 11	. . . . 5 ⊢ (¬ 𝑅 ∈ V → ∪ 𝑛 ∈ ℕ0 ∅ = ∅)
13		reldmrelexp 14918	. . . . . . 7 ⊢ Rel dom ↑𝑟
14	13	ovprc1 7401	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → (𝑅↑𝑟𝑛) = ∅)
15	14	iuneq2d 4988	. . . . 5 ⊢ (¬ 𝑅 ∈ V → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 ∅)
16		fvprc 6839	. . . . 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 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 4963	. . . . . 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 3093	. . . . 5 ⊢ (∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵 ↔ ∃𝑛 ∈ ℕ0 ⟨𝐴, 𝐵⟩ ∈ (𝑅↑𝑟𝑛))
25	21, 22, 24	3bitr4g 313	. . . 4 ⊢ (𝜑 → (𝐴∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵))
26	19, 25	sylan9bb 510	. . 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 686	. 2 ⊢ (𝜑 → (𝐴((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵))
28		df-rtrclrec 14953	. . 3 ⊢ t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
29		fveq1 6846	. . . . . 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 230	1 ⊢ (𝜑 → (𝐴(t*rec‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3069  Vcvv 3446  ∅c0 4287  ⟨cop 4597  ∪ ciun 4959   class class class wbr 5110   ↦ cmpt 5193  Rel wrel 5643  ‘cfv 6501  (class class class)co 7362  ℕ₀cn0 12422  ↑_𝑟crelexp 14916  t*reccrtrcl 14952

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 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-1cn 11118  ax-addcl 11120

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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-nn 12163  df-n0 12423  df-relexp 14917  df-rtrclrec 14953

This theorem is referenced by:  rtrclreclem3  14957  rtrclind  14962