Theorem ssiun2s 2594

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-17 971	. . 3 |- (y e. C -> A.x y e. C)
2		ax-17 971	. . . 4 |- (C e. A -> A.x C e. A)
3		ax-17 971	. . . . 5 |- (y e. D -> A.x y e. D)
4		hbiu1 2584	. . . . 5 |- (y e. U_x e. A B -> A.x y e. U_x e. A B)
5	3, 4	hbss 2062	. . . 4 |- (D (_ U_x e. A B -> A.x D (_ U_x e. A B)
6	2, 5	hbim 1007	. . 3 |- ((C e. A -> D (_ U_x e. A B) -> A.x(C e. A -> D (_ U_x e. A B))
7		eleq1 1534	. . . 4 |- (x = C -> (x e. A <-> C e. A))
8		ssiun2s.1	. . . . 5 |- (x = C -> B = D)
9	8	sseq1d 2088	. . . 4 |- (x = C -> (B (_ U_x e. A B <-> D (_ U_x e. A B))
10	7, 9	imbi12d 626	. . 3 |- (x = C -> ((x e. A -> B (_ U_x e. A B) <-> (C e. A -> D (_ U_x e. A B)))
11		ssiun2 2593	. . 3 |- (x e. A -> B (_ U_x e. A B)
12	1, 6, 10, 11	vtoclgf 1846	. 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 956   e. wcel 958   (_ wss 2047  U_ciun 2566

This theorem is referenced by:  oaordi 4180  omordi 4197  alephordlem2 4873  alephordi 4874

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rex 1650  df-v 1812  df-in 2051  df-ss 2053  df-iun 2568

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