Theorem brrtrclrec 40062

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 11904	. 2 ⊢ ℕ0 ∈ V
2		brrtrclrec.def	. . 3 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
3	2	briunov2 40047	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ ℕ0 ∈ V) → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ ℕ0 𝑋(𝑅↑𝑟𝑛)𝑌))
4	1, 3	mpan2 689	1 ⊢ (𝑅 ∈ 𝑉 → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ ℕ0 𝑋(𝑅↑𝑟𝑛)𝑌))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1537   ∈ wcel 2114  ∃wrex 3139  Vcvv 3494  ∪ ciun 4919   class class class wbr 5066   ↦ cmpt 5146  ‘cfv 6355  (class class class)co 7156  ℕ₀cn0 11898  ↑_𝑟crelexp 14379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-1cn 10595  ax-addcl 10597

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  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 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-om 7581  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-nn 11639  df-n0 11899

This theorem is referenced by: (None)