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Theorem iunconst 5001
Description: Indexed union of a constant class, i.e. where 𝐵 does not depend on 𝑥. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iunconst (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iunconst
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eliun 4995 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
2 r19.9rzv 4500 . . 3 (𝐴 ≠ ∅ → (𝑦𝐵 ↔ ∃𝑥𝐴 𝑦𝐵))
31, 2bitr4id 290 . 2 (𝐴 ≠ ∅ → (𝑦 𝑥𝐴 𝐵𝑦𝐵))
43eqrdv 2735 1 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wne 2940  wrex 3070  c0 4333   ciun 4991
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-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3482  df-dif 3954  df-nul 4334  df-iun 4993
This theorem is referenced by:  iununi  5099  oe1m  8583  oarec  8600  oelim2  8633  bnj1143  34804  poimirlem32  37659  mblfinlem2  37665  scottrankd  44267  hoicvr  46563  ovnlecvr2  46625  iunhoiioo  46691
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