Theorem hbrel 3245

Description: Bound-variable hypothesis builder for a relation.

Hypothesis

Ref	Expression
hbrel.1	|- (y e. A -> A.x y e. A)

Assertion

Ref	Expression
hbrel	|- (Rel A -> A.xRel A)

Distinct variable groups:   y,A   x,y

Proof of Theorem hbrel

Step	Hyp	Ref	Expression
1		hbrel.1	. . 3 |- (y e. A -> A.x y e. A)
2		ax-17 971	. . 3 |- (y e. (V X. V) -> A.x y e. (V X. V))
3	1, 2	hbss 2062	. 2 |- (A (_ (V X. V) -> A.x A (_ (V X. V))
4		df-rel 3185	. 2 |- (Rel A <-> A (_ (V X. V))
5	4	albii 999	. 2 |- (A.xRel A <-> A.x A (_ (V X. V))
6	3, 4, 5	3imtr4 219	1 |- (Rel A -> A.xRel A)

Syntax hints:   -> wi 3  A.wal 954   e. wcel 958  Vcvv 1811   (_ wss 2047   X. cxp 3168  Rel wrel 3175

This theorem is referenced by:  hbfun 3536

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-in 2051  df-ss 2053  df-rel 3185

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