Theorem abrexex 5744

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 5386, funex 5384, fnex 5383, resfunexg 5382, and funimaexg 4983. See also abrexex2 5751. (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 2040	. . 3 |- ( x e. A |-> B ) = ( x e. A |-> B )
2	1	rnmpt 4582	. 2 |- ran ( x e. A |-> B ) = { y | E. x e. A y = B }
3		abrexex.1	. . . 4 |- A e. _V
4	3	mptex 5387	. . 3 |- ( x e. A |-> B ) e. _V
5	4	rnex 4599	. 2 |- ran ( x e. A |-> B ) e. _V
6	2, 5	eqeltrri 2111	1 |- { y | E. x e. A y = B } e. _V

Syntax hints:   = wceq 1243   e. wcel 1393   {cab 2026   E.wrex 2307   _Vcvv 2557   |-> cmpt 3818   ran crn 4346

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910

This theorem is referenced by:  ab2rexex  5758  shftfval  9422