Theorem dfrtrclrec2 15001

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 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝑅 ∈ V)
2		nn0ex 12474	. . . . . . 7 ⊢ ℕ0 ∈ V
3		ovex 7437	. . . . . . 7 ⊢ (𝑅↑𝑟𝑛) ∈ V
4	2, 3	iunex 7950	. . . . . 6 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∈ V
5		oveq1 7411	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛))
6	5	iuneq2d 5025	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
7		eqid 2733	. . . . . . 7 ⊢ (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
8	6, 7	fvmptg 6992	. . . . . 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 414	. . . 4 ⊢ (𝜑 → (𝑅 ∈ V → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)))
11		iun0 5064	. . . . . 6 ⊢ ∪ 𝑛 ∈ ℕ0 ∅ = ∅
12	11	a1i 11	. . . . 5 ⊢ (¬ 𝑅 ∈ V → ∪ 𝑛 ∈ ℕ0 ∅ = ∅)
13		reldmrelexp 14964	. . . . . . 7 ⊢ Rel dom ↑𝑟
14	13	ovprc1 7443	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → (𝑅↑𝑟𝑛) = ∅)
15	14	iuneq2d 5025	. . . . 5 ⊢ (¬ 𝑅 ∈ V → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 ∅)
16		fvprc 6880	. . . . 5 ⊢ (¬ 𝑅 ∈ V → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∅)
17	12, 15, 16	3eqtr4rd 2784	. . . 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 5149	. . . 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 5000	. . . . . 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 5148	. . . . 5 ⊢ (𝐴∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
23		df-br 5148	. . . . . 6 ⊢ (𝐴(𝑅↑𝑟𝑛)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅↑𝑟𝑛))
24	23	rexbii 3095	. . . . 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 511	. . 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 687	. 2 ⊢ (𝜑 → (𝐴((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵))
28		df-rtrclrec 14999	. . 3 ⊢ t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
29		fveq1 6887	. . . . . 6 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅))
30	29	breqd 5158	. . . . 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 344	. . . 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 341	. . 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 397   = wceq 1542   ∈ wcel 2107  ∃wrex 3071  Vcvv 3475  ∅c0 4321  ⟨cop 4633  ∪ ciun 4996   class class class wbr 5147   ↦ cmpt 5230  Rel wrel 5680  ‘cfv 6540  (class class class)co 7404  ℕ₀cn0 12468  ↑_𝑟crelexp 14962  t*reccrtrcl 14998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720  ax-cnex 11162  ax-1cn 11164  ax-addcl 11166

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-nn 12209  df-n0 12469  df-relexp 14963  df-rtrclrec 14999

This theorem is referenced by:  rtrclreclem3  15003  rtrclind  15008