Theorem reuuni4 3813

Description: Derive the property of "the unique element in A such that ph " when expressed explicitly as U.{x e. A | ph}.

Assertion

Ref	Expression
reuuni4	|- (E!x e. A ph -> [U.{x e. A | ph} / x]ph)

Distinct variable group:   x,A

Proof of Theorem reuuni4

Step	Hyp	Ref	Expression
1		reucl 3213	. 2 |- (E!x e. A ph -> U.{x e. A | ph} e. A)
2		reurex 2440	. . . 4 |- (E!x e. A ph -> E.x e. A ph)
3		hbreu1 2252	. . . . 5 |- (E!x e. A ph -> A.xE!x e. A ph)
4		hbrab1 2257	. . . . . . 7 |- (y e. {x e. A | ph} -> A.x y e. {x e. A | ph})
5	4	hbuni 3183	. . . . . 6 |- (y e. U.{x e. A | ph} -> A.x y e. U.{x e. A | ph})
6	5	hbsbc1 2462	. . . . 5 |- ((U.{x e. A | ph} e. _V -> [U.{x e. A | ph} / x]ph) -> A.x(U.{x e. A | ph} e. _V -> [U.{x e. A | ph} / x]ph))
7		reuuni1 3808	. . . . . . . . . 10 |- ((x e. A /\ E!x e. A ph) -> (ph <-> U.{x e. A | ph} = x))
8		sbceq1a 2456	. . . . . . . . . . 11 |- (x = U.{x e. A | ph} -> (ph <-> [U.{x e. A | ph} / x]ph))
9	8	eqcoms 1887	. . . . . . . . . 10 |- (U.{x e. A | ph} = x -> (ph <-> [U.{x e. A | ph} / x]ph))
10	7, 9	syl6bi 231	. . . . . . . . 9 |- ((x e. A /\ E!x e. A ph) -> (ph -> (ph <-> [U.{x e. A | ph} / x]ph)))
11	10	ibd 654	. . . . . . . 8 |- ((x e. A /\ E!x e. A ph) -> (ph -> [U.{x e. A | ph} / x]ph))
12	11	expcom 403	. . . . . . 7 |- (E!x e. A ph -> (x e. A -> (ph -> [U.{x e. A | ph} / x]ph)))
13	12	a1i 8	. . . . . 6 |- (U.{x e. A | ph} e. _V -> (E!x e. A ph -> (x e. A -> (ph -> [U.{x e. A | ph} / x]ph))))
14	13	com4l 43	. . . . 5 |- (E!x e. A ph -> (x e. A -> (ph -> (U.{x e. A | ph} e. _V -> [U.{x e. A | ph} / x]ph))))
15	3, 6, 14	r19.23ad 2213	. . . 4 |- (E!x e. A ph -> (E.x e. A ph -> (U.{x e. A | ph} e. _V -> [U.{x e. A | ph} / x]ph)))
16	2, 15	mpd 29	. . 3 |- (E!x e. A ph -> (U.{x e. A | ph} e. _V -> [U.{x e. A | ph} / x]ph))
17		elisset 2299	. . 3 |- (U.{x e. A | ph} e. A -> U.{x e. A | ph} e. _V)
18	16, 17	syl5 20	. 2 |- (E!x e. A ph -> (U.{x e. A | ph} e. A -> [U.{x e. A | ph} / x]ph))
19	1, 18	mpd 29	1 |- (E!x e. A ph -> [U.{x e. A | ph} / x]ph)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  [wsbc 1534  E.wrex 2106  E!wreu 2107  {crab 2108  _Vcvv 2292  U.cuni 3177

This theorem is referenced by:  reucl2 3814  reuuniss 3815  reuuniss2 3817  ordtypelem6 5689  ordtypelem6OLD 15380

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-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  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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