Theorem mptfng 5313

Description: The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.)

Hypothesis

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

Assertion

Ref	Expression
mptfng	|- ( A. x e. A B e. _V <-> F Fn A )

Distinct variable group:   x, A

Allowed substitution hints:   B( x)   F( x)

Proof of Theorem mptfng

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eueq 2897	. . 3 |- ( B e. _V <-> E! y y = B )
2	1	ralbii 2472	. 2 |- ( A. x e. A B e. _V <-> A. x e. A E! y y = B )
3		mptfng.1	. . . 4 |- F = ( x e. A |-> B )
4		df-mpt 4045	. . . 4 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
5	3, 4	eqtri 2186	. . 3 |- F = { <. x , y >. | ( x e. A /\ y = B ) }
6	5	fnopabg 5311	. 2 |- ( A. x e. A E! y y = B <-> F Fn A )
7	2, 6	bitri 183	1 |- ( A. x e. A B e. _V <-> F Fn A )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1343   E!weu 2014   e. wcel 2136   A.wral 2444   _Vcvv 2726   {copab 4042   |-> cmpt 4043   Fn wfn 5183

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-fun 5190  df-fn 5191

This theorem is referenced by:  fnmpt  5314  fnmpti  5316  mpteqb  5576  cc3  7209