Theorem ssiun2s 2598

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-17 973	. . 3 |- (y e. C -> A.x y e. C)
2		ax-17 973	. . . 4 |- (C e. A -> A.x C e. A)
3		ax-17 973	. . . . 5 |- (y e. D -> A.x y e. D)
4		hbiu1 2588	. . . . 5 |- (y e. U_x e. A B -> A.x y e. U_x e. A B)
5	3, 4	hbss 2065	. . . 4 |- (D (_ U_x e. A B -> A.x D (_ U_x e. A B)
6	2, 5	hbim 1009	. . 3 |- ((C e. A -> D (_ U_x e. A B) -> A.x(C e. A -> D (_ U_x e. A B))
7		eleq1 1537	. . . 4 |- (x = C -> (x e. A <-> C e. A))
8		ssiun2s.1	. . . . 5 |- (x = C -> B = D)
9	8	sseq1d 2091	. . . 4 |- (x = C -> (B (_ U_x e. A B <-> D (_ U_x e. A B))
10	7, 9	imbi12d 628	. . 3 |- (x = C -> ((x e. A -> B (_ U_x e. A B) <-> (C e. A -> D (_ U_x e. A B)))
11		ssiun2 2597	. . 3 |- (x e. A -> B (_ U_x e. A B)
12	1, 6, 10, 11	vtoclgf 1849	. 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 958   e. wcel 960   (_ wss 2050  U_ciun 2570

This theorem is referenced by:  oaordi 4186  omordi 4203  alephordlem2 4884  alephordi 4885

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-rex 1653  df-v 1815  df-in 2054  df-ss 2056  df-iun 2572

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