Theorem elrtrclrec 43671

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 12530	. 2 ⊢ ℕ0 ∈ V
2		rtrclrec.def	. . 3 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
3	2	eliunov2 43669	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ ℕ0 ∈ V) → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ ℕ0 𝑋 ∈ (𝑅↑𝑟𝑛)))
4	1, 3	mpan2 691	1 ⊢ (𝑅 ∈ 𝑉 → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ ℕ0 𝑋 ∈ (𝑅↑𝑟𝑛)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2106  ∃wrex 3068  Vcvv 3478  ∪ ciun 4996   ↦ cmpt 5231  ‘cfv 6563  (class class class)co 7431  ℕ₀cn0 12524  ↑_𝑟crelexp 15055

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-addcl 11213

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-nn 12265  df-n0 12525

This theorem is referenced by: (None)