Theorem abrexex 3845

Description: Existence of a class abstraction of existentially restricted sets. x is normally a free-variable parameter in the class expression substituted for B, which can be thought of as B(x). This simple-looking theorem is actually quite powerful and appears to involve the Axiom of Replacement in an intrinsic way, as can be seen by tracing back through the path abrexexlem2 3844, abrexexlem1 3843, fvresex 3842, resfunexg 3565, and funimaexg 3561. See also abrexex2 3856.

Hypothesis

Ref	Expression
abrexex.1	|- A e. V

Assertion

Ref	Expression
abrexex	|- {y | E.x e. A y = B} e. V

Distinct variable groups:   x,y,A   y,B

Proof of Theorem abrexex

Step	Hyp	Ref	Expression
1		abrexex.1	. . 3 |- A e. V
2		class2set 2724	. . 3 |- {z e. B | B e. V} e. V
3	1, 2	abrexexlem2 3844	. 2 |- {y | E.x e. A y = {z e. B | B e. V}} e. V
4		visset 1804	. . . . . . 7 |- y e. V
5		eleq1 1526	. . . . . . 7 |- (y = B -> (y e. V <-> B e. V))
6	4, 5	mpbii 193	. . . . . 6 |- (y = B -> B e. V)
7		ax-1 4	. . . . . . . . 9 |- (B e. V -> (z e. B -> B e. V))
8	7	r19.21aiv 1705	. . . . . . . 8 |- (B e. V -> A.z e. B B e. V)
9		rabid2 1762	. . . . . . . 8 |- (B = {z e. B | B e. V} <-> A.z e. B B e. V)
10	8, 9	sylibr 200	. . . . . . 7 |- (B e. V -> B = {z e. B | B e. V})
11	10	eqeq2d 1478	. . . . . 6 |- (B e. V -> (y = B <-> y = {z e. B | B e. V}))
12	6, 11	syl 10	. . . . 5 |- (y = B -> (y = B <-> y = {z e. B | B e. V}))
13	12	ibi 590	. . . 4 |- (y = B -> y = {z e. B | B e. V})
14	13	r19.22si 1726	. . 3 |- (E.x e. A y = B -> E.x e. A y = {z e. B | B e. V})
15	14	ss2abi 2110	. 2 |- {y | E.x e. A y = B} (_ {y | E.x e. A y = {z e. B | B e. V}}
16	3, 15	ssexi 2710	1 |- {y | E.x e. A y = B} e. V

Syntax hints:   <-> wb 146   = wceq 953   e. wcel 955  {cab 1456  A.wral 1637  E.wrex 1638  {crab 1640  Vcvv 1802

This theorem is referenced by:  abrexexg 3846  iunon 3894  oprvalex 4026  aceq5lem4 4710  aceq6b 4714  kmlem10 4746

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fv 3188

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