Theorem hbiu1 2581

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 2565	. 2 |- U_x e. A B = {z | E.x e. A z e. B}
2		hbre1 1688	. . 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 1562	1 |- (y e. U_x e. A B -> A.x y e. U_x e. A B)

Syntax hints:   -> wi 3  A.wal 953   e. wcel 957  {cab 1463  E.wrex 1645  U_ciun 2563

This theorem is referenced by:  ssiun2s 2591  ixpf 4353  r1val1 4645  rankuni2 4677  rankval4 4689  cplem2 4708

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rex 1649  df-iun 2565

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