MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reldm Structured version   Unicode version

Theorem reldm 6858
Description: An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)
Assertion
Ref Expression
reldm  |-  ( Rel 
A  ->  dom  A  =  ran  ( x  e.  A  |->  ( 1st `  x
) ) )
Distinct variable group:    x, A

Proof of Theorem reldm
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 releldm2 6857 . . 3  |-  ( Rel 
A  ->  ( y  e.  dom  A  <->  E. z  e.  A  ( 1st `  z )  =  y ) )
2 fvex 5891 . . . . . 6  |-  ( 1st `  x )  e.  _V
3 eqid 2429 . . . . . 6  |-  ( x  e.  A  |->  ( 1st `  x ) )  =  ( x  e.  A  |->  ( 1st `  x
) )
42, 3fnmpti 5724 . . . . 5  |-  ( x  e.  A  |->  ( 1st `  x ) )  Fn  A
5 fvelrnb 5928 . . . . 5  |-  ( ( x  e.  A  |->  ( 1st `  x ) )  Fn  A  -> 
( y  e.  ran  ( x  e.  A  |->  ( 1st `  x
) )  <->  E. z  e.  A  ( (
x  e.  A  |->  ( 1st `  x ) ) `  z )  =  y ) )
64, 5ax-mp 5 . . . 4  |-  ( y  e.  ran  ( x  e.  A  |->  ( 1st `  x ) )  <->  E. z  e.  A  ( (
x  e.  A  |->  ( 1st `  x ) ) `  z )  =  y )
7 fveq2 5881 . . . . . . . 8  |-  ( x  =  z  ->  ( 1st `  x )  =  ( 1st `  z
) )
8 fvex 5891 . . . . . . . 8  |-  ( 1st `  z )  e.  _V
97, 3, 8fvmpt 5964 . . . . . . 7  |-  ( z  e.  A  ->  (
( x  e.  A  |->  ( 1st `  x
) ) `  z
)  =  ( 1st `  z ) )
109eqeq1d 2431 . . . . . 6  |-  ( z  e.  A  ->  (
( ( x  e.  A  |->  ( 1st `  x
) ) `  z
)  =  y  <->  ( 1st `  z )  =  y ) )
1110rexbiia 2933 . . . . 5  |-  ( E. z  e.  A  ( ( x  e.  A  |->  ( 1st `  x
) ) `  z
)  =  y  <->  E. z  e.  A  ( 1st `  z )  =  y )
1211a1i 11 . . . 4  |-  ( Rel 
A  ->  ( E. z  e.  A  (
( x  e.  A  |->  ( 1st `  x
) ) `  z
)  =  y  <->  E. z  e.  A  ( 1st `  z )  =  y ) )
136, 12syl5rbb 261 . . 3  |-  ( Rel 
A  ->  ( E. z  e.  A  ( 1st `  z )  =  y  <->  y  e.  ran  ( x  e.  A  |->  ( 1st `  x
) ) ) )
141, 13bitrd 256 . 2  |-  ( Rel 
A  ->  ( y  e.  dom  A  <->  y  e.  ran  ( x  e.  A  |->  ( 1st `  x
) ) ) )
1514eqrdv 2426 1  |-  ( Rel 
A  ->  dom  A  =  ran  ( x  e.  A  |->  ( 1st `  x
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437    e. wcel 1870   E.wrex 2783    |-> cmpt 4484   dom cdm 4854   ran crn 4855   Rel wrel 4859    Fn wfn 5596   ` cfv 5601   1stc1st 6805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-1st 6807  df-2nd 6808
This theorem is referenced by:  fidomdm  7859  dmct  28141
  Copyright terms: Public domain W3C validator