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Theorem reldm 6871
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 6870 . . 3  |-  ( Rel 
A  ->  ( y  e.  dom  A  <->  E. z  e.  A  ( 1st `  z )  =  y ) )
2 fvex 5898 . . . . . 6  |-  ( 1st `  x )  e.  _V
3 eqid 2462 . . . . . 6  |-  ( x  e.  A  |->  ( 1st `  x ) )  =  ( x  e.  A  |->  ( 1st `  x
) )
42, 3fnmpti 5728 . . . . 5  |-  ( x  e.  A  |->  ( 1st `  x ) )  Fn  A
5 fvelrnb 5935 . . . . 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 5888 . . . . . . . 8  |-  ( x  =  z  ->  ( 1st `  x )  =  ( 1st `  z
) )
8 fvex 5898 . . . . . . . 8  |-  ( 1st `  z )  e.  _V
97, 3, 8fvmpt 5971 . . . . . . 7  |-  ( z  e.  A  ->  (
( x  e.  A  |->  ( 1st `  x
) ) `  z
)  =  ( 1st `  z ) )
109eqeq1d 2464 . . . . . 6  |-  ( z  e.  A  ->  (
( ( x  e.  A  |->  ( 1st `  x
) ) `  z
)  =  y  <->  ( 1st `  z )  =  y ) )
1110rexbiia 2900 . . . . 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 266 . . 3  |-  ( Rel 
A  ->  ( E. z  e.  A  ( 1st `  z )  =  y  <->  y  e.  ran  ( x  e.  A  |->  ( 1st `  x
) ) ) )
141, 13bitrd 261 . 2  |-  ( Rel 
A  ->  ( y  e.  dom  A  <->  y  e.  ran  ( x  e.  A  |->  ( 1st `  x
) ) ) )
1514eqrdv 2460 1  |-  ( Rel 
A  ->  dom  A  =  ran  ( x  e.  A  |->  ( 1st `  x
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1455    e. wcel 1898   E.wrex 2750    |-> cmpt 4475   dom cdm 4853   ran crn 4854   Rel wrel 4858    Fn wfn 5596   ` cfv 5601   1stc1st 6818
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-8 1900  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-pow 4595  ax-pr 4653  ax-un 6610
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-sbc 3280  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-uni 4213  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-rn 4864  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-1st 6820  df-2nd 6821
This theorem is referenced by:  fidomdm  7879  dmct  28347
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