Theorem rnmpt 4926

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 4925	. 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 4107	. . . 4 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
4	2, 3	eqtri 2226	. . 3 |- F = { <. x , y >. | ( x e. A /\ y = B ) }
5	4	rneqi 4906	. 2 |- ran F = ran { <. x , y >. | ( x e. A /\ y = B ) }
6		df-rex 2490	. . 3 |- ( E. x e. A y = B <-> E. x ( x e. A /\ y = B ) )
7	6	abbii 2321	. 2 |- { y | E. x e. A y = B } = { y | E. x ( x e. A /\ y = B ) }
8	1, 5, 7	3eqtr4i 2236	1 |- ran F = { y | E. x e. A y = B }

Syntax hints:   /\ wa 104   = wceq 1373   E.wex 1515   e. wcel 2176   {cab 2191   E.wrex 2485   {copab 4104   |-> cmpt 4105   ran crn 4676

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-mpt 4107  df-cnv 4683  df-dm 4685  df-rn 4686

This theorem is referenced by:  elrnmpt  4927  elrnmpt1  4929  elrnmptg  4930  dfiun3g  4935  dfiin3g  4936  fnrnfv  5625  fmpt  5730  fnasrn  5758  fnasrng  5760  fliftf  5868  abrexex  6202  abrexexg  6203  fo1st  6243  fo2nd  6244  qliftf  6707  negfi  11539  4sqlem11  12724  4sqlem12  12725  quslem  13156  restco  14646  2lgslem1b  15566