Theorem hbop 2494

Description: Bound-variable hypothesis builder for ordered pairs.

Hypotheses

Ref	Expression
hbop.1	|- (y e. A -> A.x y e. A)
hbop.2	|- (y e. B -> A.x y e. B)

Assertion

Ref	Expression
hbop	|- (y e. <.A, B>. -> A.x y e. <.A, B>.)

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

Proof of Theorem hbop

Step	Hyp	Ref	Expression
1		df-op 2414	. 2 |- <.A, B>. = {{A}, {A, B}}
2		hbop.1	. . . 4 |- (y e. A -> A.x y e. A)
3	2	hbsn 2436	. . 3 |- (y e. {A} -> A.x y e. {A})
4		hbop.2	. . . 4 |- (y e. B -> A.x y e. B)
5	2, 4	hbpr 2424	. . 3 |- (y e. {A, B} -> A.x y e. {A, B})
6	3, 5	hbpr 2424	. 2 |- (y e. {{A}, {A, B}} -> A.x y e. {{A}, {A, B}})
7	1, 6	hbxfr 1562	1 |- (y e. <.A, B>. -> A.x y e. <.A, B>.)

Syntax hints:   -> wi 3  A.wal 953   e. wcel 957  {csn 2407  {cpr 2408  <.cop 2409

This theorem is referenced by:  hbopd 2495  hbbr 2655  moop2 2798  hbima 3408  hbopr 3978  xpmapenlem1 4489  seq1lem1 6264

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-or 224  df-an 225  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1810  df-un 2048  df-sn 2410  df-pr 2411  df-op 2414

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