Theorem abrexex 6117

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 5741, funex 5739, fnex 5738, resfunexg 5737, and funimaexg 5300. See also abrexex2 6124. (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 2177	. . 3 |- ( x e. A |-> B ) = ( x e. A |-> B )
2	1	rnmpt 4875	. 2 |- ran ( x e. A |-> B ) = { y | E. x e. A y = B }
3		abrexex.1	. . . 4 |- A e. _V
4	3	mptex 5742	. . 3 |- ( x e. A |-> B ) e. _V
5	4	rnex 4894	. 2 |- ran ( x e. A |-> B ) e. _V
6	2, 5	eqeltrri 2251	1 |- { y | E. x e. A y = B } e. _V

Syntax hints:   = wceq 1353   e. wcel 2148   {cab 2163   E.wrex 2456   _Vcvv 2737   |-> cmpt 4064   ran crn 4627

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224

This theorem is referenced by:  ab2rexex  6131  shftfval  10829