Theorem rnmpt 4782

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 4781	. 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 3986	. . . 4 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
4	2, 3	eqtri 2158	. . 3 |- F = { <. x , y >. | ( x e. A /\ y = B ) }
5	4	rneqi 4762	. 2 |- ran F = ran { <. x , y >. | ( x e. A /\ y = B ) }
6		df-rex 2420	. . 3 |- ( E. x e. A y = B <-> E. x ( x e. A /\ y = B ) )
7	6	abbii 2253	. 2 |- { y | E. x e. A y = B } = { y | E. x ( x e. A /\ y = B ) }
8	1, 5, 7	3eqtr4i 2168	1 |- ran F = { y | E. x e. A y = B }

Syntax hints:   /\ wa 103   = wceq 1331   E.wex 1468   e. wcel 1480   {cab 2123   E.wrex 2415   {copab 3983   |-> cmpt 3984   ran crn 4535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-mpt 3986  df-cnv 4542  df-dm 4544  df-rn 4545

This theorem is referenced by:  elrnmpt  4783  elrnmpt1  4785  elrnmptg  4786  dfiun3g  4791  dfiin3g  4792  fnrnfv  5461  fmpt  5563  fnasrn  5591  fnasrng  5593  fliftf  5693  abrexex  6008  abrexexg  6009  fo1st  6048  fo2nd  6049  qliftf  6507  negfi  10992  restco  12332