Theorem ssiun2s 3960

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 2339	. 2 |- F/_ x C
2		nfcv 2339	. . 3 |- F/_ x D
3		nfiu1 3946	. . 3 |- F/_ x U_ x e. A B
4	2, 3	nfss 3176	. 2 |- F/ x D C_ U_ x e. A B
5		ssiun2s.1	. . 3 |- ( x = C -> B = D )
6	5	sseq1d 3212	. 2 |- ( x = C -> ( B C_ U_ x e. A B <-> D C_ U_ x e. A B ) )
7		ssiun2 3959	. 2 |- ( x e. A -> B C_ U_ x e. A B )
8	1, 4, 6, 7	vtoclgaf 2829	1 |- ( C e. A -> D C_ U_ x e. A B )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   C_ wss 3157   U_ciun 3916

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-in 3163  df-ss 3170  df-iun 3918

This theorem is referenced by:  imasaddvallemg  12958