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Theorem briunov2uz 42284
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. The index set 𝑁 is restricted to an upper range of integers. (Contributed by RP, 2-Jun-2020.)
Hypothesis
Ref Expression
briunov2uz.def 𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))
Assertion
Ref Expression
briunov2uz ((𝑅𝑈𝑁 = (ℤ𝑀)) → (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)   𝑀(𝑛,𝑟)   𝑋(𝑟)   𝑌(𝑟)

Proof of Theorem briunov2uz
StepHypRef Expression
1 simpr 485 . . 3 ((𝑅𝑈𝑁 = (ℤ𝑀)) → 𝑁 = (ℤ𝑀))
2 fvex 6892 . . 3 (ℤ𝑀) ∈ V
31, 2eqeltrdi 2841 . 2 ((𝑅𝑈𝑁 = (ℤ𝑀)) → 𝑁 ∈ V)
4 briunov2uz.def . . 3 𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))
54briunov2 42268 . 2 ((𝑅𝑈𝑁 ∈ V) → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛𝑁 𝑋(𝑅 𝑛)𝑌))
63, 5syldan 591 1 ((𝑅𝑈𝑁 = (ℤ𝑀)) → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛𝑁 𝑋(𝑅 𝑛)𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wrex 3070  Vcvv 3474   ciun 4991   class class class wbr 5142  cmpt 5225  cfv 6533  (class class class)co 7394  cuz 12806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-iota 6485  df-fun 6535  df-fv 6541  df-ov 7397
This theorem is referenced by:  iunrelexpuztr  42305
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