Theorem dfdm3 3308

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 3194	. 2 |- dom A = {x | E.y xAy}
2		df-br 2625	. . . 4 |- (xAy <-> <.x, y>. e. A)
3	2	exbii 1053	. . 3 |- (E.y xAy <-> E.y<.x, y>. e. A)
4	3	abbii 1578	. 2 |- {x | E.y xAy} = {x | E.y<.x, y>. e. A}
5	1, 4	eqtr 1498	1 |- dom A = {x | E.y<.x, y>. e. A}

Syntax hints:   = wceq 958   e. wcel 960  E.wex 982  {cab 1466  <.cop 2415   class class class wbr 2624  dom cdm 3176

This theorem is referenced by:  dfdmf 3312  dm0 3329  dmsn0 3330  dmsnsn0 3331  dmsnop 3334

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-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-br 2625  df-dm 3194

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