Theorem ss2iun 4536

Description: Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
ss2iun	|- ( A. x e. A B C_ C -> U_ x e. A B C_ U_ x e. A C )

Proof of Theorem ss2iun

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3597	. . . . 5 |- ( B C_ C -> ( y e. B -> y e. C ) )
2	1	ralimi 2952	. . . 4 |- ( A. x e. A B C_ C -> A. x e. A ( y e. B -> y e. C ) )
3		rexim 3008	. . . 4 |- ( A. x e. A ( y e. B -> y e. C ) -> ( E. x e. A y e. B -> E. x e. A y e. C ) )
4	2, 3	syl 17	. . 3 |- ( A. x e. A B C_ C -> ( E. x e. A y e. B -> E. x e. A y e. C ) )
5		eliun 4524	. . 3 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
6		eliun 4524	. . 3 |- ( y e. U_ x e. A C <-> E. x e. A y e. C )
7	4, 5, 6	3imtr4g 285	. 2 |- ( A. x e. A B C_ C -> ( y e. U_ x e. A B -> y e. U_ x e. A C ) )
8	7	ssrdv 3609	1 |- ( A. x e. A B C_ C -> U_ x e. A B C_ U_ x e. A C )

Syntax hints:   -> wi 4   e. wcel 1990   A.wral 2912   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:  iuneq2  4537  abnexg  6964  oawordri  7630  omwordri  7652  oewordri  7672  oeworde  7673  r1val1  8649  cfslb2n  9090  imasaddvallem  16189  dprdss  18428  tgcmp  21204  txcmplem1  21444  txcmplem2  21445  xkococnlem  21462  alexsubALT  21855  ptcmplem3  21858  metnrmlem2  22663  uniiccvol  23348  dvfval  23661  bnj1145  31061  bnj1136  31065  filnetlem3  32375  poimirlem32  33441  sstotbnd2  33573  equivtotbnd  33577  trclrelexplem  38003  corcltrcl  38031  cotrclrcl  38034  ovolval5lem2  40867  ovolval5lem3  40868  smflimsuplem7  41032