Theorem abrexex 4836

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 4835, abrexexlem1 4834, fvresex 4833, resfunexg 4500, and funimaexg 4495. See also abrexex2 4847.

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 3471	. . 3 |- {z e. B | B e. _V} e. _V
3	1, 2	abrexexlem2 4835	. 2 |- {y | E.x e. A y = {z e. B | B e. _V}} e. _V
4		visset 2295	. . . . . . 7 |- y e. _V
5		eleq1 1957	. . . . . . 7 |- (y = B -> (y e. _V <-> B e. _V))
6	4, 5	mpbii 210	. . . . . 6 |- (y = B -> B e. _V)
7		ax-1 4	. . . . . . . . 9 |- (B e. _V -> (z e. B -> B e. _V))
8	7	r19.21aiv 2175	. . . . . . . 8 |- (B e. _V -> A.z e. B B e. _V)
9		rabid2 2254	. . . . . . . 8 |- (B = {z e. B | B e. _V} <-> A.z e. B B e. _V)
10	8, 9	sylibr 217	. . . . . . 7 |- (B e. _V -> B = {z e. B | B e. _V})
11	10	eqeq2d 1895	. . . . . 6 |- (B e. _V -> (y = B <-> y = {z e. B | B e. _V}))
12	6, 11	syl 12	. . . . 5 |- (y = B -> (y = B <-> y = {z e. B | B e. _V}))
13	12	ibi 652	. . . 4 |- (y = B -> y = {z e. B | B e. _V})
14	13	reximi 2198	. . 3 |- (E.x e. A y = B -> E.x e. A y = {z e. B | B e. _V})
15	14	ss2abi 2679	. 2 |- {y | E.x e. A y = B} C_ {y | E.x e. A y = {z e. B | B e. _V}}
16	3, 15	ssexi 3456	1 |- {y | E.x e. A y = B} e. _V

Syntax hints:   <-> wb 163   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  E.wrex 2106  {crab 2108  _Vcvv 2292

This theorem is referenced by:  abrexexg 4837  ab2rexex 4971  oprvexOLDOLD 4974  iunon 5114  hartog 5693  aceq5lem4 5900  aceq6b 5904  kmlem10 5936  txbas 8933  cncomp 10331  bintop 14901  fictblem 15370  fictb 15371  hartogOLD 15384  compsublem 15430  compsub 15431  hscptsscld 15434  pointset 17222

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-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  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-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014

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