Theorem elrtrclrec 40024

Description: Membership in the indexed union of relation exponentiation over the natural numbers (including zero) is equivalent to the existence of at least one number such that the element is a member of that relationship power. (Contributed by RP, 2-Jun-2020.)

Hypothesis

Ref	Expression
rtrclrec.def	⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))

Assertion

Ref	Expression
elrtrclrec	⊢ (𝑅 ∈ 𝑉 → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ ℕ0 𝑋 ∈ (𝑅↑𝑟𝑛)))

Distinct variable groups:   𝑛,𝑟,𝐶   𝑅,𝑛,𝑟   𝑛,𝑋

Allowed substitution hints:   𝑉(𝑛,𝑟)   𝑋(𝑟)

Proof of Theorem elrtrclrec

Step	Hyp	Ref	Expression
1		nn0ex 11902	. 2 ⊢ ℕ0 ∈ V
2		rtrclrec.def	. . 3 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
3	2	eliunov2 40022	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ ℕ0 ∈ V) → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ ℕ0 𝑋 ∈ (𝑅↑𝑟𝑛)))
4	1, 3	mpan2 689	1 ⊢ (𝑅 ∈ 𝑉 → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ ℕ0 𝑋 ∈ (𝑅↑𝑟𝑛)))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1533   ∈ wcel 2110  ∃wrex 3139  Vcvv 3494  ∪ ciun 4918   ↦ cmpt 5145  ‘cfv 6354  (class class class)co 7155  ℕ₀cn0 11896  ↑_𝑟crelexp 14378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-1cn 10594  ax-addcl 10596

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-om 7580  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-nn 11638  df-n0 11897

This theorem is referenced by: (None)