Theorem iunss1 4532

Description: Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iunss1	|- ( A C_ B -> U_ x e. A C C_ U_ x e. B 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.

Step	Hyp	Ref	Expression
1		ssrexv 3667	. . 3 |- ( A C_ B -> ( E. x e. A y e. C -> E. x e. B y e. C ) )
2		eliun 4524	. . 3 |- ( y e. U_ x e. A C <-> E. x e. A y e. C )
3		eliun 4524	. . 3 |- ( y e. U_ x e. B C <-> E. x e. B y e. C )
4	1, 2, 3	3imtr4g 285	. 2 |- ( A C_ B -> ( y e. U_ x e. A C -> y e. U_ x e. B C ) )
5	4	ssrdv 3609	1 |- ( A C_ B -> U_ x e. A C C_ U_ x e. B C )

Syntax hints:   -> wi 4   e. wcel 1990   E.wrex 2913   C_ wss 3574   U_ciun 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-iun 4522

This theorem is referenced by:  iuneq1  4534  iunxdif2  4568  oelim2  7675  fsumiun  14553  ovolfiniun  23269  uniioovol  23347  fusgreghash2wspv  27199  esum2dlem  30154  esum2d  30155  carsgclctunlem2  30381  bnj1413  31103  bnj1408  31104  volsupnfl  33454  corclrcl  37999  cotrcltrcl  38017  iuneqfzuzlem  39550  fsumiunss  39807  sge0iunmptlemfi  40630  sge0iunmptlemre  40632  carageniuncllem1  40735  carageniuncllem2  40736  caratheodorylem2  40741  ovnsubaddlem1  40784