Theorem dmopab3 3322

Description: The domain of a restricted class of ordered pairs.

Assertion

Ref	Expression
dmopab3	|- (A.x e. A E.yph <-> dom {<.x, y>. | (x e. A /\ ph)} = A)

Distinct variable group:   x,y,A

Proof of Theorem dmopab3

Step	Hyp	Ref	Expression
1		df-ral 1649	. 2 |- (A.x e. A E.yph <-> A.x(x e. A -> E.yph))
2		pm4.71 635	. . 3 |- ((x e. A -> E.yph) <-> (x e. A <-> (x e. A /\ E.yph)))
3	2	albii 999	. 2 |- (A.x(x e. A -> E.yph) <-> A.x(x e. A <-> (x e. A /\ E.yph)))
4		dmopab 3320	. . . . 5 |- dom {<.x, y>. | (x e. A /\ ph)} = {x | E.y(x e. A /\ ph)}
5		19.42v 1308	. . . . . 6 |- (E.y(x e. A /\ ph) <-> (x e. A /\ E.yph))
6	5	abbii 1575	. . . . 5 |- {x | E.y(x e. A /\ ph)} = {x | (x e. A /\ E.yph)}
7	4, 6	eqtr 1495	. . . 4 |- dom {<.x, y>. | (x e. A /\ ph)} = {x | (x e. A /\ E.yph)}
8	7	eqeq1i 1482	. . 3 |- (dom {<.x, y>. | (x e. A /\ ph)} = A <-> {x | (x e. A /\ E.yph)} = A)
9		eqcom 1477	. . 3 |- (A = {x | (x e. A /\ E.yph)} <-> {x | (x e. A /\ E.yph)} = A)
10		abeq2 1568	. . 3 |- (A = {x | (x e. A /\ E.yph)} <-> A.x(x e. A <-> (x e. A /\ E.yph)))
11	8, 9, 10	3bitr2r 180	. 2 |- (A.x(x e. A <-> (x e. A /\ E.yph)) <-> dom {<.x, y>. | (x e. A /\ ph)} = A)
12	1, 3, 11	3bitr 177	1 |- (A.x e. A E.yph <-> dom {<.x, y>. | (x e. A /\ ph)} = A)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  A.wral 1645  {copab 2666  dom cdm 3170

This theorem is referenced by:  dmxp 3332  fnopabg 3615  fopab2 3823  dmrecpq 5074

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-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  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  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-dm 3188

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