Theorem ssiun2s 2589

Description: Subset relationship for an indexed union.

Hypothesis

Ref	Expression
ssiun2s.1	|- (x = C -> B = D)

Assertion

Ref	Expression
ssiun2s	|- (C e. A -> D (_ U_x e. A B)

Distinct variable groups:   x,A   x,C   x,D

Proof of Theorem ssiun2s

Step	Hyp	Ref	Expression
1		ax-17 969	. . 3 |- (y e. C -> A.x y e. C)
2		ax-17 969	. . . 4 |- (C e. A -> A.x C e. A)
3		ax-17 969	. . . . 5 |- (y e. D -> A.x y e. D)
4		hbiu1 2579	. . . . 5 |- (y e. U_x e. A B -> A.x y e. U_x e. A B)
5	3, 4	hbss 2058	. . . 4 |- (D (_ U_x e. A B -> A.x D (_ U_x e. A B)
6	2, 5	hbim 1005	. . 3 |- ((C e. A -> D (_ U_x e. A B) -> A.x(C e. A -> D (_ U_x e. A B))
7		eleq1 1531	. . . 4 |- (x = C -> (x e. A <-> C e. A))
8		ssiun2s.1	. . . . 5 |- (x = C -> B = D)
9	8	sseq1d 2084	. . . 4 |- (x = C -> (B (_ U_x e. A B <-> D (_ U_x e. A B))
10	7, 9	imbi12d 625	. . 3 |- (x = C -> ((x e. A -> B (_ U_x e. A B) <-> (C e. A -> D (_ U_x e. A B)))
11		ssiun2 2588	. . 3 |- (x e. A -> B (_ U_x e. A B)
12	1, 6, 10, 11	vtoclgf 1842	. 2 |- (C e. A -> (C e. A -> D (_ U_x e. A B))
13	12	pm2.43i 64	1 |- (C e. A -> D (_ U_x e. A B)

Syntax hints:   -> wi 3   = wceq 954   e. wcel 956   (_ wss 2043  U_ciun 2561

This theorem is referenced by:  oaordi 4170  omordi 4187  alephordlem2 4853  alephordi 4854

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-rex 1647  df-v 1808  df-in 2047  df-ss 2049  df-iun 2563

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