Theorem ssiun2s 3932

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 2319	. 2 |- F/_ x C
2		nfcv 2319	. . 3 |- F/_ x D
3		nfiu1 3918	. . 3 |- F/_ x U_ x e. A B
4	2, 3	nfss 3150	. 2 |- F/ x D C_ U_ x e. A B
5		ssiun2s.1	. . 3 |- ( x = C -> B = D )
6	5	sseq1d 3186	. 2 |- ( x = C -> ( B C_ U_ x e. A B <-> D C_ U_ x e. A B ) )
7		ssiun2 3931	. 2 |- ( x e. A -> B C_ U_ x e. A B )
8	1, 4, 6, 7	vtoclgaf 2804	1 |- ( C e. A -> D C_ U_ x e. A B )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   C_ wss 3131   U_ciun 3888

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-in 3137  df-ss 3144  df-iun 3890

This theorem is referenced by:  imasaddvallemg  12742