Theorem ssiun2s 3971

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 2348	. 2 |- F/_ x C
2		nfcv 2348	. . 3 |- F/_ x D
3		nfiu1 3957	. . 3 |- F/_ x U_ x e. A B
4	2, 3	nfss 3186	. 2 |- F/ x D C_ U_ x e. A B
5		ssiun2s.1	. . 3 |- ( x = C -> B = D )
6	5	sseq1d 3222	. 2 |- ( x = C -> ( B C_ U_ x e. A B <-> D C_ U_ x e. A B ) )
7		ssiun2 3970	. 2 |- ( x e. A -> B C_ U_ x e. A B )
8	1, 4, 6, 7	vtoclgaf 2838	1 |- ( C e. A -> D C_ U_ x e. A B )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   C_ wss 3166   U_ciun 3927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-in 3172  df-ss 3179  df-iun 3929

This theorem is referenced by:  imasaddvallemg  13147