Theorem rnmpt 4896

Description: The range of a function in maps-to notation. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
rnmpt.1	|- F = ( x e. A |-> B )

Assertion

Ref	Expression
rnmpt	|- ran F = { y | E. x e. A y = B }

Distinct variable groups:   y, A   y, B   x, y

Allowed substitution hints:   A( x)   B( x)   F( x, y)

Proof of Theorem rnmpt

Step	Hyp	Ref	Expression
1		rnopab 4895	. 2 |- ran { <. x , y >. | ( x e. A /\ y = B ) } = { y | E. x ( x e. A /\ y = B ) }
2		rnmpt.1	. . . 4 |- F = ( x e. A |-> B )
3		df-mpt 4084	. . . 4 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
4	2, 3	eqtri 2210	. . 3 |- F = { <. x , y >. | ( x e. A /\ y = B ) }
5	4	rneqi 4876	. 2 |- ran F = ran { <. x , y >. | ( x e. A /\ y = B ) }
6		df-rex 2474	. . 3 |- ( E. x e. A y = B <-> E. x ( x e. A /\ y = B ) )
7	6	abbii 2305	. 2 |- { y | E. x e. A y = B } = { y | E. x ( x e. A /\ y = B ) }
8	1, 5, 7	3eqtr4i 2220	1 |- ran F = { y | E. x e. A y = B }

Syntax hints:   /\ wa 104   = wceq 1364   E.wex 1503   e. wcel 2160   {cab 2175   E.wrex 2469   {copab 4081   |-> cmpt 4082   ran crn 4648

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-br 4022  df-opab 4083  df-mpt 4084  df-cnv 4655  df-dm 4657  df-rn 4658

This theorem is referenced by:  elrnmpt  4897  elrnmpt1  4899  elrnmptg  4900  dfiun3g  4905  dfiin3g  4906  fnrnfv  5586  fmpt  5690  fnasrn  5718  fnasrng  5720  fliftf  5824  abrexex  6146  abrexexg  6147  fo1st  6186  fo2nd  6187  qliftf  6650  negfi  11277  4sqlem11  12444  4sqlem12  12445  quslem  12812  restco  14159