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Theorem iunss1 4666
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 3816 . . 3 (𝐴𝐵 → (∃𝑥𝐴 𝑦𝐶 → ∃𝑥𝐵 𝑦𝐶))
2 eliun 4658 . . 3 (𝑦 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑦𝐶)
3 eliun 4658 . . 3 (𝑦 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝑦𝐶)
41, 2, 33imtr4g 285 . 2 (𝐴𝐵 → (𝑦 𝑥𝐴 𝐶𝑦 𝑥𝐵 𝐶))
54ssrdv 3758 1 (𝐴𝐵 𝑥𝐴 𝐶 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145  wrex 3062  wss 3723   ciun 4654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-in 3730  df-ss 3737  df-iun 4656
This theorem is referenced by:  iuneq1  4668  iunxdif2  4702  oelim2  7827  fsumiun  14753  ovolfiniun  23482  uniioovol  23560  fusgreghash2wspv  27510  esum2dlem  30487  esum2d  30488  carsgclctunlem2  30714  bnj1413  31434  bnj1408  31435  volsupnfl  33780  corclrcl  38518  cotrcltrcl  38536  iuneqfzuzlem  40059  fsumiunss  40318  sge0iunmptlemfi  41140  sge0iunmptlemre  41142  carageniuncllem1  41248  carageniuncllem2  41249  caratheodorylem2  41254  ovnsubaddlem1  41297
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