Theorem mptfng 5380

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 2932	. . 3 |- ( B e. _V <-> E! y y = B )
2	1	ralbii 2500	. 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 4093	. . . 4 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
5	3, 4	eqtri 2214	. . 3 |- F = { <. x , y >. | ( x e. A /\ y = B ) }
6	5	fnopabg 5378	. 2 |- ( A. x e. A E! y y = B <-> F Fn A )
7	2, 6	bitri 184	1 |- ( A. x e. A B e. _V <-> F Fn A )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   E!weu 2042   e. wcel 2164   A.wral 2472   _Vcvv 2760   {copab 4090   |-> cmpt 4091   Fn wfn 5250

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-fun 5257  df-fn 5258

This theorem is referenced by:  fnmpt  5381  fnmpti  5383  mpteqb  5649  cc3  7330