Theorem ssiun2s 4564

Description: Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.)

Hypothesis

Ref	Expression
ssiun2s.1	|- ( x = C -> B = D )

Assertion

Ref	Expression
ssiun2s	|- ( C e. A -> D C_ U_ x e. A B )

Distinct variable groups:   x, A   x, C   x, D

Allowed substitution hint:   B( x)

Proof of Theorem ssiun2s

Step	Hyp	Ref	Expression
1		nfcv 2764	. 2 |- F/_ x C
2		nfcv 2764	. . 3 |- F/_ x D
3		nfiu1 4550	. . 3 |- F/_ x U_ x e. A B
4	2, 3	nfss 3596	. 2 |- F/ x D C_ U_ x e. A B
5		ssiun2s.1	. . 3 |- ( x = C -> B = D )
6	5	sseq1d 3632	. 2 |- ( x = C -> ( B C_ U_ x e. A B <-> D C_ U_ x e. A B ) )
7		ssiun2 4563	. 2 |- ( x e. A -> B C_ U_ x e. A B )
8	1, 4, 6, 7	vtoclgaf 3271	1 |- ( C e. A -> D C_ U_ x e. A B )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   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:  onfununi  7438  oaordi  7626  omordi  7646  dffi3  8337  alephordi  8897  domtriomlem  9264  pwxpndom2  9487  wunex2  9560  imasaddvallem  16189  imasvscaval  16198  iundisj2  23317  voliunlem1  23318  volsup  23324  iundisj2fi  29556  bnj906  31000  bnj1137  31063  bnj1408  31104  cvmliftlem10  31276  cvmliftlem13  31278  sstotbnd2  33573  mapdrvallem3  36935  fvmptiunrelexplb0d  37976  fvmptiunrelexplb1d  37978  corclrcl  37999  trclrelexplem  38003  corcltrcl  38031  cotrclrcl  38034  iunincfi  39272  iundjiunlem  40676  caratheodorylem1  40740  ovnhoilem1  40815