Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brrtrclrec Structured version   Visualization version   GIF version

Theorem brrtrclrec 42433
Description: Two classes related by 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 two classes are related by that relationship power. (Contributed by RP, 2-Jun-2020.)
Hypothesis
Ref Expression
brrtrclrec.def 𝐶 = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
Assertion
Ref Expression
brrtrclrec (𝑅𝑉 → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛 ∈ ℕ0 𝑋(𝑅𝑟𝑛)𝑌))
Distinct variable groups:   𝑛,𝑟,𝐶   𝑅,𝑛,𝑟   𝑛,𝑋   𝑛,𝑌
Allowed substitution hints:   𝑉(𝑛,𝑟)   𝑋(𝑟)   𝑌(𝑟)

Proof of Theorem brrtrclrec
StepHypRef Expression
1 nn0ex 12474 . 2 0 ∈ V
2 brrtrclrec.def . . 3 𝐶 = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
32briunov2 42418 . 2 ((𝑅𝑉 ∧ ℕ0 ∈ V) → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛 ∈ ℕ0 𝑋(𝑅𝑟𝑛)𝑌))
41, 3mpan2 689 1 (𝑅𝑉 → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛 ∈ ℕ0 𝑋(𝑅𝑟𝑛)𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2106  wrex 3070  Vcvv 3474   ciun 4996   class class class wbr 5147  cmpt 5230  cfv 6540  (class class class)co 7405  0cn0 12468  𝑟crelexp 14962
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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-1cn 11164  ax-addcl 11166
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  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 7408  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-nn 12209  df-n0 12469
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator