Theorem briunov2 44391

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 44388	. 2 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (⟨𝑋, 𝑌⟩ ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 ⟨𝑋, 𝑌⟩ ∈ (𝑅 ↑ 𝑛)))
3		df-br 5111	. 2 ⊢ (𝑋(𝐶‘𝑅)𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ (𝐶‘𝑅))
4		df-br 5111	. . 3 ⊢ (𝑋(𝑅 ↑ 𝑛)𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ (𝑅 ↑ 𝑛))
5	4	rexbii 3112	. 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))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  Vcvv 3455  ⟨cop 4596  ∪ ciun 4957   class class class wbr 5110   ↦ cmpt 5193  ‘cfv 6538  (class class class)co 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7415

This theorem is referenced by:  brmptiunrelexpd  44392  brtrclrec  44405  brrtrclrec  44406  briunov2uz  44407