Theorem dmopab3 3279

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 1625	. 2 |- (A.x e. A E.yph <-> A.x(x e. A -> E.yph))
2		pm4.71 633	. . 3 |- ((x e. A -> E.yph) <-> (x e. A <-> (x e. A /\ E.yph)))
3	2	albii 975	. 2 |- (A.x(x e. A -> E.yph) <-> A.x(x e. A <-> (x e. A /\ E.yph)))
4		dmopab 3277	. . . . 5 |- dom {<.x, y>. | (x e. A /\ ph)} = {x | E.y(x e. A /\ ph)}
5		19.42v 1290	. . . . . 6 |- (E.y(x e. A /\ ph) <-> (x e. A /\ E.yph))
6	5	abbii 1551	. . . . 5 |- {x | E.y(x e. A /\ ph)} = {x | (x e. A /\ E.yph)}
7	4, 6	eqtr 1471	. . . 4 |- dom {<.x, y>. | (x e. A /\ ph)} = {x | (x e. A /\ E.yph)}
8	7	eqeq1i 1458	. . 3 |- (dom {<.x, y>. | (x e. A /\ ph)} = A <-> {x | (x e. A /\ E.yph)} = A)
9		eqcom 1453	. . 3 |- (A = {x | (x e. A /\ E.yph)} <-> {x | (x e. A /\ E.yph)} = A)
10		abeq2 1544	. . 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 950  E.wex 956   = wceq 1099   e. wcel 1105  {cab 1440  A.wral 1621  {copab 2634  dom cdm 3133

This theorem is referenced by:  dmxp 3291  fnopabg 3555  fopab2 3762  dmrecpq 4997

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671  ax-pow 2710  ax-pr 2747

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-op 2387  df-br 2588  df-opab 2635  df-dm 3151

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