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Theorem briunov2 43695
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 ↦ 𝑛𝑁 (𝑟 𝑛))
Assertion
Ref Expression
briunov2 ((𝑅𝑈𝑁𝑉) → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛𝑁 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
StepHypRef Expression
1 briunov2.def . . 3 𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))
21eliunov2 43692 . 2 ((𝑅𝑈𝑁𝑉) → (⟨𝑋, 𝑌⟩ ∈ (𝐶𝑅) ↔ ∃𝑛𝑁𝑋, 𝑌⟩ ∈ (𝑅 𝑛)))
3 df-br 5144 . 2 (𝑋(𝐶𝑅)𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ (𝐶𝑅))
4 df-br 5144 . . 3 (𝑋(𝑅 𝑛)𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ (𝑅 𝑛))
54rexbii 3094 . 2 (∃𝑛𝑁 𝑋(𝑅 𝑛)𝑌 ↔ ∃𝑛𝑁𝑋, 𝑌⟩ ∈ (𝑅 𝑛))
62, 3, 53bitr4g 314 1 ((𝑅𝑈𝑁𝑉) → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛𝑁 𝑋(𝑅 𝑛)𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  cop 4632   ciun 4991   class class class wbr 5143  cmpt 5225  cfv 6561  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434
This theorem is referenced by:  brmptiunrelexpd  43696  brtrclrec  43709  brrtrclrec  43710  briunov2uz  43711
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