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Theorem iunss1 4498
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 3646 . . 3 (𝐴𝐵 → (∃𝑥𝐴 𝑦𝐶 → ∃𝑥𝐵 𝑦𝐶))
2 eliun 4490 . . 3 (𝑦 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑦𝐶)
3 eliun 4490 . . 3 (𝑦 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝑦𝐶)
41, 2, 33imtr4g 285 . 2 (𝐴𝐵 → (𝑦 𝑥𝐴 𝐶𝑦 𝑥𝐵 𝐶))
54ssrdv 3589 1 (𝐴𝐵 𝑥𝐴 𝐶 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  wrex 2908  wss 3555   ciun 4485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-in 3562  df-ss 3569  df-iun 4487
This theorem is referenced by:  iuneq1  4500  iunxdif2  4534  oelim2  7620  fsumiun  14480  ovolfiniun  23176  uniioovol  23253  fusgreghash2wspv  27057  esum2dlem  29935  esum2d  29936  carsgclctunlem2  30162  bnj1413  30811  bnj1408  30812  volsupnfl  33086  corclrcl  37480  cotrcltrcl  37498  iuneqfzuzlem  39014  fsumiunss  39211  sge0iunmptlemfi  39937  sge0iunmptlemre  39939  carageniuncllem1  40042  carageniuncllem2  40043  caratheodorylem2  40048  ovnsubaddlem1  40091
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