Theorem hbbr 2663

Description: Bound-variable hypothesis builder for binary relation.

Hypotheses

Ref	Expression
hbbr.1	|- (y e. A -> A.x y e. A)
hbbr.2	|- (y e. R -> A.x y e. R)
hbbr.3	|- (y e. B -> A.x y e. B)

Assertion

Ref	Expression
hbbr	|- (ARB -> A.x ARB)

Distinct variable groups:   y,A   y,B   y,R   x,y

Proof of Theorem hbbr

Step	Hyp	Ref	Expression
1		hbbr.1	. . . 4 |- (y e. A -> A.x y e. A)
2		hbbr.3	. . . 4 |- (y e. B -> A.x y e. B)
3	1, 2	hbop 2500	. . 3 |- (y e. <.A, B>. -> A.x y e. <.A, B>.)
4		hbbr.2	. . 3 |- (y e. R -> A.x y e. R)
5	3, 4	hbel 1569	. 2 |- (<.A, B>. e. R -> A.x<.A, B>. e. R)
6		df-br 2625	. 2 |- (ARB <-> <.A, B>. e. R)
7	6	albii 1001	. 2 |- (A.x ARB <-> A.x<.A, B>. e. R)
8	5, 6, 7	3imtr4 219	1 |- (ARB -> A.x ARB)

Syntax hints:   -> wi 3  A.wal 956   e. wcel 960  <.cop 2415   class class class wbr 2624

This theorem is referenced by:  hbbrd 2664  sbcbrg 2667  hbco 3293  hbcnv 3301  dffunmof 3536  funfv2f 3778  hbiso 3898  uniimadomf 4821  ondomcard 4868  cardmin 4871  alephordlem1 4883  lble 6049  hbsum1 6983  hbsum 6984  isumcmpi 7215  irredt 10317

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625

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