Theorem dfdm3 3302

Description: Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24.

Assertion

Ref	Expression
dfdm3	|- dom A = {x | E.y<.x, y>. e. A}

Distinct variable group:   x,y,A

Proof of Theorem dfdm3

Step	Hyp	Ref	Expression
1		df-dm 3188	. 2 |- dom A = {x | E.y xAy}
2		df-br 2620	. . . 4 |- (xAy <-> <.x, y>. e. A)
3	2	exbii 1051	. . 3 |- (E.y xAy <-> E.y<.x, y>. e. A)
4	3	abbii 1575	. 2 |- {x | E.y xAy} = {x | E.y<.x, y>. e. A}
5	1, 4	eqtr 1495	1 |- dom A = {x | E.y<.x, y>. e. A}

Syntax hints:   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  <.cop 2411   class class class wbr 2619  dom cdm 3170

This theorem is referenced by:  dfdmf 3306  dm0 3323  dmsn0 3324  dmsnsn0 3325  dmsnop 3328

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-br 2620  df-dm 3188

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