Theorem dmopabss 3318

Description: Upper bound for the domain of a restricted class of ordered pairs.

Assertion

Ref	Expression
dmopabss	|- dom {<.x, y>. | (x e. A /\ ph)} (_ A

Distinct variable group:   x,y,A

Proof of Theorem dmopabss

Step	Hyp	Ref	Expression
1		dmopab 3317	. 2 |- dom {<.x, y>. | (x e. A /\ ph)} = {x | E.y(x e. A /\ ph)}
2		19.42v 1308	. . . 4 |- (E.y(x e. A /\ ph) <-> (x e. A /\ E.yph))
3	2	abbii 1574	. . 3 |- {x | E.y(x e. A /\ ph)} = {x | (x e. A /\ E.yph)}
4		ssab2 2128	. . 3 |- {x | (x e. A /\ E.yph)} (_ A
5	3, 4	eqsstr 2089	. 2 |- {x | E.y(x e. A /\ ph)} (_ A
6	1, 5	eqsstr 2089	1 |- dom {<.x, y>. | (x e. A /\ ph)} (_ A

Syntax hints:   /\ wa 223   e. wcel 957  E.wex 979  {cab 1463   (_ wss 2045  {copab 2663  dom cdm 3167

This theorem is referenced by:  opabex 3606  opabex2g 3608  fvopab4ndm 3781  uzssz 6380  dmadjss 9810

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-br 2617  df-opab 2664  df-dm 3185

Copyright terms: Public domain