Theorem eldm2g 3309

Description: Domain membership. Theorem 4 of [Suppes] p. 59.

Assertion

Ref	Expression
eldm2g	|- (A e. C -> (A e. dom B <-> E.y<.A, y>. e. B))

Distinct variable groups:   y,A   y,B

Proof of Theorem eldm2g

Step	Hyp	Ref	Expression
1		eleq1 1534	. 2 |- (x = A -> (x e. dom B <-> A e. dom B))
2		opeq1 2487	. . . 4 |- (x = A -> <.x, y>. = <.A, y>.)
3	2	eleq1d 1540	. . 3 |- (x = A -> (<.x, y>. e. B <-> <.A, y>. e. B))
4	3	exbidv 1279	. 2 |- (x = A -> (E.y<.x, y>. e. B <-> E.y<.A, y>. e. B))
5		visset 1813	. . 3 |- x e. V
6	5	eldm2 3308	. 2 |- (x e. dom B <-> E.y<.x, y>. e. B)
7	1, 4, 6	vtoclbg 1848	1 |- (A e. C -> (A e. dom B <-> E.y<.A, y>. e. B))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  E.wex 980  <.cop 2411  dom cdm 3170

This theorem is referenced by:  dmfco 3773

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  df-br 2620  df-dm 3188

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