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Theorem reldm 5840
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 5839 . . 3  |-  ( Rel 
A  ->  ( y  e.  dom  A  <->  E. z  e.  A  ( 1st `  z )  =  y ) )
2 vex 2577 . . . . . . 7  |-  x  e. 
_V
3 1stexg 5822 . . . . . . 7  |-  ( x  e.  _V  ->  ( 1st `  x )  e. 
_V )
42, 3ax-mp 7 . . . . . 6  |-  ( 1st `  x )  e.  _V
5 eqid 2056 . . . . . 6  |-  ( x  e.  A  |->  ( 1st `  x ) )  =  ( x  e.  A  |->  ( 1st `  x
) )
64, 5fnmpti 5055 . . . . 5  |-  ( x  e.  A  |->  ( 1st `  x ) )  Fn  A
7 fvelrnb 5249 . . . . 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 ) )
86, 7ax-mp 7 . . . 4  |-  ( y  e.  ran  ( x  e.  A  |->  ( 1st `  x ) )  <->  E. z  e.  A  ( (
x  e.  A  |->  ( 1st `  x ) ) `  z )  =  y )
9 fveq2 5206 . . . . . . . 8  |-  ( x  =  z  ->  ( 1st `  x )  =  ( 1st `  z
) )
10 vex 2577 . . . . . . . . 9  |-  z  e. 
_V
11 1stexg 5822 . . . . . . . . 9  |-  ( z  e.  _V  ->  ( 1st `  z )  e. 
_V )
1210, 11ax-mp 7 . . . . . . . 8  |-  ( 1st `  z )  e.  _V
139, 5, 12fvmpt 5277 . . . . . . 7  |-  ( z  e.  A  ->  (
( x  e.  A  |->  ( 1st `  x
) ) `  z
)  =  ( 1st `  z ) )
1413eqeq1d 2064 . . . . . 6  |-  ( z  e.  A  ->  (
( ( x  e.  A  |->  ( 1st `  x
) ) `  z
)  =  y  <->  ( 1st `  z )  =  y ) )
1514rexbiia 2356 . . . . 5  |-  ( E. z  e.  A  ( ( x  e.  A  |->  ( 1st `  x
) ) `  z
)  =  y  <->  E. z  e.  A  ( 1st `  z )  =  y )
1615a1i 9 . . . 4  |-  ( Rel 
A  ->  ( E. z  e.  A  (
( x  e.  A  |->  ( 1st `  x
) ) `  z
)  =  y  <->  E. z  e.  A  ( 1st `  z )  =  y ) )
178, 16syl5rbb 186 . . 3  |-  ( Rel 
A  ->  ( E. z  e.  A  ( 1st `  z )  =  y  <->  y  e.  ran  ( x  e.  A  |->  ( 1st `  x
) ) ) )
181, 17bitrd 181 . 2  |-  ( Rel 
A  ->  ( y  e.  dom  A  <->  y  e.  ran  ( x  e.  A  |->  ( 1st `  x
) ) ) )
1918eqrdv 2054 1  |-  ( Rel 
A  ->  dom  A  =  ran  ( x  e.  A  |->  ( 1st `  x
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102    = wceq 1259    e. wcel 1409   E.wrex 2324   _Vcvv 2574    |-> cmpt 3846   dom cdm 4373   ran crn 4374   Rel wrel 4378    Fn wfn 4925   ` cfv 4930   1stc1st 5793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fo 4936  df-fv 4938  df-1st 5795  df-2nd 5796
This theorem is referenced by: (None)
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