Theorem rnmpt 4945

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 4944	. 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 4123	. . . 4 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
4	2, 3	eqtri 2228	. . 3 |- F = { <. x , y >. | ( x e. A /\ y = B ) }
5	4	rneqi 4925	. 2 |- ran F = ran { <. x , y >. | ( x e. A /\ y = B ) }
6		df-rex 2492	. . 3 |- ( E. x e. A y = B <-> E. x ( x e. A /\ y = B ) )
7	6	abbii 2323	. 2 |- { y | E. x e. A y = B } = { y | E. x ( x e. A /\ y = B ) }
8	1, 5, 7	3eqtr4i 2238	1 |- ran F = { y | E. x e. A y = B }

Syntax hints:   /\ wa 104   = wceq 1373   E.wex 1516   e. wcel 2178   {cab 2193   E.wrex 2487   {copab 4120   |-> cmpt 4121   ran crn 4694

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-mpt 4123  df-cnv 4701  df-dm 4703  df-rn 4704

This theorem is referenced by:  elrnmpt  4946  elrnmpt1  4948  elrnmptg  4949  dfiun3g  4954  dfiin3g  4955  fnrnfv  5648  fmpt  5753  fnasrn  5781  fnasrng  5783  fliftf  5891  abrexex  6225  abrexexg  6226  fo1st  6266  fo2nd  6267  qliftf  6730  negfi  11654  4sqlem11  12839  4sqlem12  12840  quslem  13271  restco  14761  2lgslem1b  15681