Theorem ssiun2s 3910

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 2308	. 2 |- F/_ x C
2		nfcv 2308	. . 3 |- F/_ x D
3		nfiu1 3896	. . 3 |- F/_ x U_ x e. A B
4	2, 3	nfss 3135	. 2 |- F/ x D C_ U_ x e. A B
5		ssiun2s.1	. . 3 |- ( x = C -> B = D )
6	5	sseq1d 3171	. 2 |- ( x = C -> ( B C_ U_ x e. A B <-> D C_ U_ x e. A B ) )
7		ssiun2 3909	. 2 |- ( x e. A -> B C_ U_ x e. A B )
8	1, 4, 6, 7	vtoclgaf 2791	1 |- ( C e. A -> D C_ U_ x e. A B )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   C_ wss 3116   U_ciun 3866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-in 3122  df-ss 3129  df-iun 3868

This theorem is referenced by: (None)