Theorem iunss1 2574

Description: Subclass theorem for indexed union.

Assertion

Ref	Expression
iunss1	|- (A (_ B -> U_x e. A C (_ U_x e. B C)

Distinct variable groups:   x,A   x,B

Proof of Theorem iunss1

Step	Hyp	Ref	Expression
1		ssel 2063	. . . . . 6 |- (A (_ B -> (x e. A -> x e. B))
2	1	anim1d 560	. . . . 5 |- (A (_ B -> ((x e. A /\ y e. C) -> (x e. B /\ y e. C)))
3	2	r19.22dv2 1736	. . . 4 |- (A (_ B -> (E.x e. A y e. C -> E.x e. B y e. C))
4	3	19.21aiv 1286	. . 3 |- (A (_ B -> A.y(E.x e. A y e. C -> E.x e. B y e. C))
5		ss2ab 2116	. . 3 |- ({y | E.x e. A y e. C} (_ {y | E.x e. B y e. C} <-> A.y(E.x e. A y e. C -> E.x e. B y e. C))
6	4, 5	sylibr 200	. 2 |- (A (_ B -> {y | E.x e. A y e. C} (_ {y | E.x e. B y e. C})
7		df-iun 2568	. 2 |- U_x e. A C = {y | E.x e. A y e. C}
8		df-iun 2568	. 2 |- U_x e. B C = {y | E.x e. B y e. C}
9	6, 7, 8	3sstr4g 2102	1 |- (A (_ B -> U_x e. A C (_ U_x e. B C)

Syntax hints:   -> wi 3  A.wal 954   e. wcel 958  {cab 1463  E.wrex 1646   (_ wss 2047  U_ciun 2566

This theorem is referenced by:  iuneq1 2575  iunxdif2 2598  oelim2 4222

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-in 2051  df-ss 2053  df-iun 2568

Copyright terms: Public domain