Theorem brrtrclrec 44641

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 12581	. 2 ⊢ ℕ0 ∈ V
2		brrtrclrec.def	. . 3 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
3	2	briunov2 44626	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ ℕ0 ∈ V) → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ ℕ0 𝑋(𝑅↑𝑟𝑛)𝑌))
4	1, 3	mpan2 704	1 ⊢ (𝑅 ∈ 𝑉 → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ ℕ0 𝑋(𝑅↑𝑟𝑛)𝑌))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  Vcvv 3450  ∪ ciun 4950   class class class wbr 5102   ↦ cmpt 5185  ‘cfv 6527  (class class class)co 7408  ℕ₀cn0 12575  ↑_𝑟crelexp 15139

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7734  ax-cnex 11227  ax-1cn 11229  ax-addcl 11231

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-nn 12305  df-n0 12576

This theorem is used by: (None)