Theorem ss2iun 2577

Description: Subclass theorem for indexed union.

Assertion

Ref	Expression
ss2iun	|- (A.x e. A B (_ C -> U_x e. A B (_ U_x e. A C)

Proof of Theorem ss2iun

Step	Hyp	Ref	Expression
1		hbra1 1687	. . . 4 |- (A.x e. A B (_ C -> A.xA.x e. A B (_ C)
2		ra4 1694	. . . . 5 |- (A.x e. A B (_ C -> (x e. A -> B (_ C))
3		ssel 2063	. . . . 5 |- (B (_ C -> (y e. B -> y e. C))
4	2, 3	syl6 22	. . . 4 |- (A.x e. A B (_ C -> (x e. A -> (y e. B -> y e. C)))
5	1, 4	r19.22d 1735	. . 3 |- (A.x e. A B (_ C -> (E.x e. A y e. B -> E.x e. A y e. C))
6		eliun 2570	. . 3 |- (y e. U_x e. A B <-> E.x e. A y e. B)
7		eliun 2570	. . 3 |- (y e. U_x e. A C <-> E.x e. A y e. C)
8	5, 6, 7	3imtr4g 553	. 2 |- (A.x e. A B (_ C -> (y e. U_x e. A B -> y e. U_x e. A C))
9	8	ssrdv 2070	1 |- (A.x e. A B (_ C -> U_x e. A B (_ U_x e. A C)

Syntax hints:   -> wi 3   e. wcel 958  A.wral 1645  E.wrex 1646   (_ wss 2047  U_ciun 2566

This theorem is referenced by:  iuneq2 2578  oawordri 4184  omwordri 4203  oewordri 4219  oeworde 4220  r1val1 4658

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-ral 1649  df-rex 1650  df-v 1812  df-in 2051  df-ss 2053  df-iun 2568

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