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Theorem eltrclrec 39461
 Description: Membership in the indexed union of relation exponentiation over the natural numbers 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
trclrec.def 𝐶 = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ (𝑟𝑟𝑛))
Assertion
Ref Expression
eltrclrec (𝑅𝑉 → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛 ∈ ℕ 𝑋 ∈ (𝑅𝑟𝑛)))
Distinct variable groups:   𝑛,𝑟,𝐶   𝑅,𝑛,𝑟   𝑛,𝑋
Allowed substitution hints:   𝑉(𝑛,𝑟)   𝑋(𝑟)

Proof of Theorem eltrclrec
StepHypRef Expression
1 nnex 11481 . 2 ℕ ∈ V
2 trclrec.def . . 3 𝐶 = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ (𝑟𝑟𝑛))
32eliunov2 39460 . 2 ((𝑅𝑉 ∧ ℕ ∈ V) → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛 ∈ ℕ 𝑋 ∈ (𝑅𝑟𝑛)))
41, 3mpan2 687 1 (𝑅𝑉 → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛 ∈ ℕ 𝑋 ∈ (𝑅𝑟𝑛)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   = wceq 1520   ∈ wcel 2079  ∃wrex 3104  Vcvv 3432  ∪ ciun 4819   ↦ cmpt 5035  ‘cfv 6217  (class class class)co 7007  ℕcn 11475  ↑𝑟crelexp 14201 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310  ax-cnex 10428  ax-1cn 10430  ax-addcl 10432 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-ov 7010  df-om 7428  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-nn 11476 This theorem is referenced by: (None)
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