Theorem abrexex 6132

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 5754, funex 5752, fnex 5751, resfunexg 5750, and funimaexg 5312. See also abrexex2 6139. (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 2187	. . 3 |- ( x e. A |-> B ) = ( x e. A |-> B )
2	1	rnmpt 4887	. 2 |- ran ( x e. A |-> B ) = { y | E. x e. A y = B }
3		abrexex.1	. . . 4 |- A e. _V
4	3	mptex 5755	. . 3 |- ( x e. A |-> B ) e. _V
5	4	rnex 4906	. 2 |- ran ( x e. A |-> B ) e. _V
6	2, 5	eqeltrri 2261	1 |- { y | E. x e. A y = B } e. _V

Syntax hints:   = wceq 1363   e. wcel 2158   {cab 2173   E.wrex 2466   _Vcvv 2749   |-> cmpt 4076   ran crn 4639

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236

This theorem is referenced by:  ab2rexex  6146  shftfval  10844