Theorem dfid4 5026

Description: The identity function using maps-to notation. (Contributed by Scott Fenton, 15-Dec-2017.)

Assertion

Ref	Expression
dfid4	|- _I = ( x e. _V |-> x )

Proof of Theorem dfid4

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equcom 1945	. . . 4 |- ( x = y <-> y = x )
2		vex 3203	. . . . 5 |- x e. _V
3	2	biantrur 527	. . . 4 |- ( y = x <-> ( x e. _V /\ y = x ) )
4	1, 3	bitri 264	. . 3 |- ( x = y <-> ( x e. _V /\ y = x ) )
5	4	opabbii 4717	. 2 |- { <. x , y >. | x = y } = { <. x , y >. | ( x e. _V /\ y = x ) }
6		df-id 5024	. 2 |- _I = { <. x , y >. | x = y }
7		df-mpt 4730	. 2 |- ( x e. _V |-> x ) = { <. x , y >. | ( x e. _V /\ y = x ) }
8	5, 6, 7	3eqtr4i 2654	1 |- _I = ( x e. _V |-> x )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {copab 4712   |-> cmpt 4729   _I cid 5023

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-opab 4713  df-mpt 4730  df-id 5024

This theorem is referenced by:  dfid5  13767  dfid6  13768  dfid7  37919