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Theorem briunov2 43672
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 43669 . 2 ((𝑅𝑈𝑁𝑉) → (⟨𝑋, 𝑌⟩ ∈ (𝐶𝑅) ↔ ∃𝑛𝑁𝑋, 𝑌⟩ ∈ (𝑅 𝑛)))
3 df-br 5149 . 2 (𝑋(𝐶𝑅)𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ (𝐶𝑅))
4 df-br 5149 . . 3 (𝑋(𝑅 𝑛)𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ (𝑅 𝑛))
54rexbii 3092 . 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 1537  wcel 2106  wrex 3068  Vcvv 3478  cop 4637   ciun 4996   class class class wbr 5148  cmpt 5231  cfv 6563  (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 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434
This theorem is referenced by:  brmptiunrelexpd  43673  brtrclrec  43686  brrtrclrec  43687  briunov2uz  43688
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