Theorem rnmpt 4857

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 4856	. 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 4050	. . . 4 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
4	2, 3	eqtri 2191	. . 3 |- F = { <. x , y >. | ( x e. A /\ y = B ) }
5	4	rneqi 4837	. 2 |- ran F = ran { <. x , y >. | ( x e. A /\ y = B ) }
6		df-rex 2454	. . 3 |- ( E. x e. A y = B <-> E. x ( x e. A /\ y = B ) )
7	6	abbii 2286	. 2 |- { y | E. x e. A y = B } = { y | E. x ( x e. A /\ y = B ) }
8	1, 5, 7	3eqtr4i 2201	1 |- ran F = { y | E. x e. A y = B }

Syntax hints:   /\ wa 103   = wceq 1348   E.wex 1485   e. wcel 2141   {cab 2156   E.wrex 2449   {copab 4047   |-> cmpt 4048   ran crn 4610

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-mpt 4050  df-cnv 4617  df-dm 4619  df-rn 4620

This theorem is referenced by:  elrnmpt  4858  elrnmpt1  4860  elrnmptg  4861  dfiun3g  4866  dfiin3g  4867  fnrnfv  5541  fmpt  5644  fnasrn  5672  fnasrng  5674  fliftf  5776  abrexex  6094  abrexexg  6095  fo1st  6134  fo2nd  6135  qliftf  6596  negfi  11184  restco  12933