Theorem hbop 2496

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 2416	. 2 |- <.A, B>. = {{A}, {A, B}}
2		hbop.1	. . . 4 |- (y e. A -> A.x y e. A)
3	2	hbsn 2438	. . 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 2426	. . 3 |- (y e. {A, B} -> A.x y e. {A, B})
6	3, 5	hbpr 2426	. 2 |- (y e. {{A}, {A, B}} -> A.x y e. {{A}, {A, B}})
7	1, 6	hbxfr 1563	1 |- (y e. <.A, B>. -> A.x y e. <.A, B>.)

Syntax hints:   -> wi 3  A.wal 954   e. wcel 958  {csn 2409  {cpr 2410  <.cop 2411

This theorem is referenced by:  hbopd 2497  hbbr 2658  moop2 2801  hbima 3411  hbopr 3981  xpmapenlem1 4496  seq1lem1 6309

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-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-op 2416

Copyright terms: Public domain