Theorem reuuni1 3808

Description: A way to express "the unique element such that" (restricted quantifier version). (The proof was shortened by Andrew Salmon, 12-Aug-2011.)

Assertion

Ref	Expression
reuuni1	|- ((x e. A /\ E!x e. A ph) -> (ph <-> U.{x e. A | ph} = x))

Proof of Theorem reuuni1

Step	Hyp	Ref	Expression
1		ibar 705	. . . 4 |- (x e. A -> (ph <-> (x e. A /\ ph)))
2		df-rab 2112	. . . . . . 7 |- {x e. A | ph} = {x | (x e. A /\ ph)}
3	2	unieqi 3187	. . . . . 6 |- U.{x e. A | ph} = U.{x | (x e. A /\ ph)}
4	3	eqeq1i 1891	. . . . 5 |- (U.{x e. A | ph} = x <-> U.{x | (x e. A /\ ph)} = x)
5	4	a1i 8	. . . 4 |- (x e. A -> (U.{x e. A | ph} = x <-> U.{x | (x e. A /\ ph)} = x))
6	1, 5	bibi12d 691	. . 3 |- (x e. A -> ((ph <-> U.{x e. A | ph} = x) <-> ((x e. A /\ ph) <-> U.{x | (x e. A /\ ph)} = x)))
7		df-reu 2111	. . . 4 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
8		euuni 3807	. . . 4 |- (E!x(x e. A /\ ph) -> ((x e. A /\ ph) <-> U.{x | (x e. A /\ ph)} = x))
9	7, 8	sylbi 216	. . 3 |- (E!x e. A ph -> ((x e. A /\ ph) <-> U.{x | (x e. A /\ ph)} = x))
10	6, 9	syl5cbir 228	. 2 |- (E!x e. A ph -> (x e. A -> (ph <-> U.{x e. A | ph} = x)))
11	10	impcom 378	1 |- ((x e. A /\ E!x e. A ph) -> (ph <-> U.{x e. A | ph} = x))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E!weu 1771  {cab 1871  E!wreu 2107  {crab 2108  U.cuni 3177

This theorem is referenced by:  reuuni2f 3810  reuuni4 3813  replim 8011  cnid 9435  mulid 9440  hilid 10661

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-uni 3178

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