Theorem abrexex 6278

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 mptexg 5878, funex 5876, fnex 5875, resfunexg 5874, and funimaexg 5414. See also abrexex2 6285. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

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

Allowed substitution hint:   B( x)

Proof of Theorem abrexex

Step	Hyp	Ref	Expression
1		eqid 2231	. . 3 |- ( x e. A |-> B ) = ( x e. A |-> B )
2	1	rnmpt 4980	. 2 |- ran ( x e. A |-> B ) = { y | E. x e. A y = B }
3		abrexex.1	. . . 4 |- A e. _V
4	3	mptex 5879	. . 3 |- ( x e. A |-> B ) e. _V
5	4	rnex 5000	. 2 |- ran ( x e. A |-> B ) e. _V
6	2, 5	eqeltrri 2305	1 |- { y | E. x e. A y = B } e. _V

Syntax hints:   = wceq 1397   e. wcel 2202   {cab 2217   E.wrex 2511   _Vcvv 2802   |-> cmpt 4150   ran crn 4726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334

This theorem is referenced by:  ab2rexex  6292  shftfval  11381