Theorem brrtrclrec 42937

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 (𝑟↑𝑟𝑛))

Assertion

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

Step	Hyp	Ref	Expression
1		nn0ex 12475	. 2 ⊢ ℕ0 ∈ V
2		brrtrclrec.def	. . 3 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
3	2	briunov2 42922	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ ℕ0 ∈ V) → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ ℕ0 𝑋(𝑅↑𝑟𝑛)𝑌))
4	1, 3	mpan2 688	1 ⊢ (𝑅 ∈ 𝑉 → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ ℕ0 𝑋(𝑅↑𝑟𝑛)𝑌))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  ∃wrex 3062  Vcvv 3466  ∪ ciun 4987   class class class wbr 5138   ↦ cmpt 5221  ‘cfv 6533  (class class class)co 7401  ℕ₀cn0 12469  ↑_𝑟crelexp 14963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-1cn 11164  ax-addcl 11166

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-om 7849  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-nn 12210  df-n0 12470

This theorem is referenced by: (None)