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Theorem iunss1 5005
Description: Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iunss1 (𝐴𝐵 𝑥𝐴 𝐶 𝑥𝐵 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iunss1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssrexv 4047 . . 3 (𝐴𝐵 → (∃𝑥𝐴 𝑦𝐶 → ∃𝑥𝐵 𝑦𝐶))
2 eliun 4995 . . 3 (𝑦 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑦𝐶)
3 eliun 4995 . . 3 (𝑦 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝑦𝐶)
41, 2, 33imtr4g 296 . 2 (𝐴𝐵 → (𝑦 𝑥𝐴 𝐶𝑦 𝑥𝐵 𝐶))
54ssrdv 3984 1 (𝐴𝐵 𝑥𝐴 𝐶 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  wrex 3065  wss 3944   ciun 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-rex 3066  df-v 3471  df-in 3951  df-ss 3961  df-iun 4993
This theorem is referenced by:  iuneq1  5007  iunxdif2  5050  oelim2  8609  fsumiun  15793  ovolfiniun  25423  uniioovol  25501  fusgreghash2wspv  30138  esum2dlem  33701  esum2d  33702  carsgclctunlem2  33929  bnj1413  34656  bnj1408  34657  volsupnfl  37127  corclrcl  43109  cotrcltrcl  43127  iuneqfzuzlem  44688  fsumiunss  44935  sge0iunmptlemfi  45773  sge0iunmptlemre  45775  carageniuncllem1  45881  carageniuncllem2  45882  caratheodorylem2  45887  ovnsubaddlem1  45930
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