Theorem hbiu1 2584

Description: Bound-variable hypothesis builder for indexed union.

Assertion

Ref	Expression
hbiu1	|- (y e. U_x e. A B -> A.x y e. U_x e. A B)

Distinct variable group:   x,y

Proof of Theorem hbiu1

Step	Hyp	Ref	Expression
1		df-iun 2568	. 2 |- U_x e. A B = {z | E.x e. A z e. B}
2		hbre1 1689	. . 3 |- (E.x e. A z e. B -> A.xE.x e. A z e. B)
3	2	hbab 1467	. 2 |- (y e. {z | E.x e. A z e. B} -> A.x y e. {z | E.x e. A z e. B})
4	1, 3	hbxfr 1563	1 |- (y e. U_x e. A B -> A.x y e. U_x e. A B)

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

This theorem is referenced by:  ssiun2s 2594  ixpf 4356  r1val1 4658  rankuni2 4690  rankval4 4702  cplem2 4721

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-iun 2568

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