Theorem ssiun2s 3917

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 2312	. 2 |- F/_ x C
2		nfcv 2312	. . 3 |- F/_ x D
3		nfiu1 3903	. . 3 |- F/_ x U_ x e. A B
4	2, 3	nfss 3140	. 2 |- F/ x D C_ U_ x e. A B
5		ssiun2s.1	. . 3 |- ( x = C -> B = D )
6	5	sseq1d 3176	. 2 |- ( x = C -> ( B C_ U_ x e. A B <-> D C_ U_ x e. A B ) )
7		ssiun2 3916	. 2 |- ( x e. A -> B C_ U_ x e. A B )
8	1, 4, 6, 7	vtoclgaf 2795	1 |- ( C e. A -> D C_ U_ x e. A B )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   C_ wss 3121   U_ciun 3873

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-iun 3875

This theorem is referenced by: (None)