Theorem briunov2 43657

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 43654	. 2 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (⟨𝑋, 𝑌⟩ ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 ⟨𝑋, 𝑌⟩ ∈ (𝑅 ↑ 𝑛)))
3		df-br 5124	. 2 ⊢ (𝑋(𝐶‘𝑅)𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ (𝐶‘𝑅))
4		df-br 5124	. . 3 ⊢ (𝑋(𝑅 ↑ 𝑛)𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ (𝑅 ↑ 𝑛))
5	4	rexbii 3082	. 2 ⊢ (∃𝑛 ∈ 𝑁 𝑋(𝑅 ↑ 𝑛)𝑌 ↔ ∃𝑛 ∈ 𝑁 ⟨𝑋, 𝑌⟩ ∈ (𝑅 ↑ 𝑛))
6	2, 3, 5	3bitr4g 314	1 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ 𝑁 𝑋(𝑅 ↑ 𝑛)𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∃wrex 3059  Vcvv 3463  ⟨cop 4612  ∪ ciun 4971   class class class wbr 5123   ↦ cmpt 5205  ‘cfv 6541  (class class class)co 7413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-iota 6494  df-fun 6543  df-fv 6549  df-ov 7416

This theorem is referenced by:  brmptiunrelexpd  43658  brtrclrec  43671  brrtrclrec  43672  briunov2uz  43673