Theorem mptfng 5725

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 3222	. . 3 |- ( B e. _V <-> E! y y = B )
2	1	ralbii 2831	. 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 4477	. . . 4 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
5	3, 4	eqtri 2484	. . 3 |- F = { <. x , y >. | ( x e. A /\ y = B ) }
6	5	fnopabg 5723	. 2 |- ( A. x e. A E! y y = B <-> F Fn A )
7	2, 6	bitri 257	1 |- ( A. x e. A B e. _V <-> F Fn A )

Syntax hints:   <-> wb 189   /\ wa 375   = wceq 1455   e. wcel 1898   E!weu 2310   A.wral 2749   _Vcvv 3057   {copab 4474   |-> cmpt 4475   Fn wfn 5596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-fun 5603  df-fn 5604

This theorem is referenced by:  fnmpt  5726  fnmpti  5728  mpteqb  5987  ofmpteq  6577  bdayfo  30613  fobigcup  30716  dihf11lem  34879