Theorem briunov2 44943

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 44940	. 2 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (⟨𝑋, 𝑌⟩ ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 ⟨𝑋, 𝑌⟩ ∈ (𝑅 ↑ 𝑛)))
3		df-br 5109	. 2 ⊢ (𝑋(𝐶‘𝑅)𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ (𝐶‘𝑅))
4		df-br 5109	. . 3 ⊢ (𝑋(𝑅 ↑ 𝑛)𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ (𝑅 ↑ 𝑛))
5	4	rexbii 3113	. 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 1570   ∈ wcel 2147  ∃wrex 3090  Vcvv 3454  ⟨cop 4840  ∪ ciun 4956   class class class wbr 5108   ↦ cmpt 5192  ‘cfv 6566  (class class class)co 7451

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454

This theorem is used by:  brmptiunrelexpd  44944  brtrclrec  44957  brrtrclrec  44958  briunov2uz  44959