Theorem mptv 4476

Description: Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)

Assertion

Ref	Expression
mptv	|- ( x e. _V |-> B ) = { <. x , y >. | y = B }

Distinct variable groups:   x, y   y, B

Allowed substitution hint:   B( x)

Proof of Theorem mptv

Step	Hyp	Ref	Expression
1		df-mpt 4444	. 2 |- ( x e. _V |-> B ) = { <. x , y >. | ( x e. _V /\ y = B ) }
2		vex 3054	. . . 4 |- x e. _V
3	2	biantrur 504	. . 3 |- ( y = B <-> ( x e. _V /\ y = B ) )
4	3	opabbii 4448	. 2 |- { <. x , y >. | y = B } = { <. x , y >. | ( x e. _V /\ y = B ) }
5	1, 4	eqtr4i 2428	1 |- ( x e. _V |-> B ) = { <. x , y >. | y = B }

Syntax hints:   /\ wa 367   = wceq 1399   e. wcel 1836   _Vcvv 3051   {copab 4441   |-> cmpt 4442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374

This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-v 3053  df-opab 4443  df-mpt 4444

This theorem is referenced by:  df1st2  6807  df2nd2  6808  fsplit  6826  rankf  8147  cnmptid  20270