Theorem abrexex 5764

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 5407, funex 5405, fnex 5404, resfunexg 5403, and funimaexg 5003. See also abrexex2 5771. (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 2081	. . 3 |- ( x e. A |-> B ) = ( x e. A |-> B )
2	1	rnmpt 4600	. 2 |- ran ( x e. A |-> B ) = { y | E. x e. A y = B }
3		abrexex.1	. . . 4 |- A e. _V
4	3	mptex 5408	. . 3 |- ( x e. A |-> B ) e. _V
5	4	rnex 4617	. 2 |- ran ( x e. A |-> B ) e. _V
6	2, 5	eqeltrri 2152	1 |- { y | E. x e. A y = B } e. _V

Syntax hints:   = wceq 1284   e. wcel 1433   {cab 2067   E.wrex 2349   _Vcvv 2601   |-> cmpt 3839   ran crn 4364

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930

This theorem is referenced by:  ab2rexex  5778  shftfval  9709