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Theorem iunss1 4832
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 3950 . . 3 (𝐴𝐵 → (∃𝑥𝐴 𝑦𝐶 → ∃𝑥𝐵 𝑦𝐶))
2 eliun 4823 . . 3 (𝑦 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑦𝐶)
3 eliun 4823 . . 3 (𝑦 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝑦𝐶)
41, 2, 33imtr4g 297 . 2 (𝐴𝐵 → (𝑦 𝑥𝐴 𝐶𝑦 𝑥𝐵 𝐶))
54ssrdv 3890 1 (𝐴𝐵 𝑥𝐴 𝐶 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2079  wrex 3104  wss 3854   ciun 4819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-ext 2767
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-v 3434  df-in 3861  df-ss 3869  df-iun 4821
This theorem is referenced by:  iuneq1  4834  iunxdif2  4870  oelim2  8062  fsumiun  14997  ovolfiniun  23773  uniioovol  23851  fusgreghash2wspv  27794  esum2dlem  30924  esum2d  30925  carsgclctunlem2  31150  bnj1413  31877  bnj1408  31878  volsupnfl  34414  corclrcl  39488  cotrcltrcl  39506  iuneqfzuzlem  41096  fsumiunss  41352  sge0iunmptlemfi  42191  sge0iunmptlemre  42193  carageniuncllem1  42299  carageniuncllem2  42300  caratheodorylem2  42305  ovnsubaddlem1  42348
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