Theorem briunov2 44789

Description: Two classes related by the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the two classes are related by that operator value. (Contributed by RP, 1-Jun-2020.)

Hypothesis

Ref	Expression
briunov2.def	⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))

Assertion

Ref	Expression
briunov2	⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ 𝑁 𝑋(𝑅 ↑ 𝑛)𝑌))

Distinct variable groups:   𝑛,𝑟,𝐶,𝑁, ↑   𝑅,𝑛,𝑟   𝑛,𝑋   𝑛,𝑌

Allowed substitution hints:   𝑈(𝑛, 𝑟)   𝑉(𝑛, 𝑟)   𝑋(𝑟)   𝑌(𝑟)

Proof of Theorem briunov2

Step	Hyp	Ref	Expression
1		briunov2.def	. . 3 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))
2	1	eliunov2 44786	. 2 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (⟨𝑋, 𝑌⟩ ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 ⟨𝑋, 𝑌⟩ ∈ (𝑅 ↑ 𝑛)))
3		df-br 5103	. 2 ⊢ (𝑋(𝐶‘𝑅)𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ (𝐶‘𝑅))
4		df-br 5103	. . 3 ⊢ (𝑋(𝑅 ↑ 𝑛)𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ (𝑅 ↑ 𝑛))
5	4	rexbii 3109	. 2 ⊢ (∃𝑛 ∈ 𝑁 𝑋(𝑅 ↑ 𝑛)𝑌 ↔ ∃𝑛 ∈ 𝑁 ⟨𝑋, 𝑌⟩ ∈ (𝑅 ↑ 𝑛))
6	2, 3, 5	3bitr4g 317	1 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ 𝑁 𝑋(𝑅 ↑ 𝑛)𝑌))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  Vcvv 3450  ⟨cop 4834  ∪ ciun 4950   class class class wbr 5102   ↦ cmpt 5185  ‘cfv 6541  (class class class)co 7422

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6496  df-fun 6543  df-fv 6549  df-ov 7425

This theorem is used by:  brmptiunrelexpd  44790  brtrclrec  44803  brrtrclrec  44804  briunov2uz  44805