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Theorem eliunov2uz 43193
Description: Membership in 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 element is a member of that operator value. The index set 𝑁 is restricted to an upper range of integers. (Contributed by RP, 2-Jun-2020.)
Hypothesis
Ref Expression
eliunov2uz.def 𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))
Assertion
Ref Expression
eliunov2uz ((𝑅𝑈𝑁 = (ℤ𝑀)) → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛𝑁 𝑋 ∈ (𝑅 𝑛)))
Distinct variable groups:   𝑛,𝑟,𝐶,𝑁,   𝑅,𝑛,𝑟   𝑛,𝑋
Allowed substitution hints:   𝑈(𝑛,𝑟)   𝑀(𝑛,𝑟)   𝑋(𝑟)

Proof of Theorem eliunov2uz
StepHypRef Expression
1 simpr 483 . . 3 ((𝑅𝑈𝑁 = (ℤ𝑀)) → 𝑁 = (ℤ𝑀))
2 fvex 6904 . . 3 (ℤ𝑀) ∈ V
31, 2eqeltrdi 2833 . 2 ((𝑅𝑈𝑁 = (ℤ𝑀)) → 𝑁 ∈ V)
4 eliunov2uz.def . . 3 𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))
54eliunov2 43173 . 2 ((𝑅𝑈𝑁 ∈ V) → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛𝑁 𝑋 ∈ (𝑅 𝑛)))
63, 5syldan 589 1 ((𝑅𝑈𝑁 = (ℤ𝑀)) → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛𝑁 𝑋 ∈ (𝑅 𝑛)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wrex 3060  Vcvv 3463   ciun 4991  cmpt 5226  cfv 6542  (class class class)co 7415  cuz 12850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7418
This theorem is referenced by: (None)
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