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Theorem reldm 6171
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 6170 . . 3  |-  ( Rel 
A  ->  ( y  e.  dom  A  <->  E. z  e.  A  ( 1st `  z )  =  y ) )
2 fvex 5539 . . . . . 6  |-  ( 1st `  x )  e.  _V
3 eqid 2283 . . . . . 6  |-  ( x  e.  A  |->  ( 1st `  x ) )  =  ( x  e.  A  |->  ( 1st `  x
) )
42, 3fnmpti 5372 . . . . 5  |-  ( x  e.  A  |->  ( 1st `  x ) )  Fn  A
5 fvelrnb 5570 . . . . 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 8 . . . 4  |-  ( y  e.  ran  ( x  e.  A  |->  ( 1st `  x ) )  <->  E. z  e.  A  ( (
x  e.  A  |->  ( 1st `  x ) ) `  z )  =  y )
7 fveq2 5525 . . . . . . . 8  |-  ( x  =  z  ->  ( 1st `  x )  =  ( 1st `  z
) )
8 fvex 5539 . . . . . . . 8  |-  ( 1st `  z )  e.  _V
97, 3, 8fvmpt 5602 . . . . . . 7  |-  ( z  e.  A  ->  (
( x  e.  A  |->  ( 1st `  x
) ) `  z
)  =  ( 1st `  z ) )
109eqeq1d 2291 . . . . . 6  |-  ( z  e.  A  ->  (
( ( x  e.  A  |->  ( 1st `  x
) ) `  z
)  =  y  <->  ( 1st `  z )  =  y ) )
1110rexbiia 2576 . . . . 5  |-  ( E. z  e.  A  ( ( x  e.  A  |->  ( 1st `  x
) ) `  z
)  =  y  <->  E. z  e.  A  ( 1st `  z )  =  y )
1211a1i 10 . . . 4  |-  ( Rel 
A  ->  ( E. z  e.  A  (
( x  e.  A  |->  ( 1st `  x
) ) `  z
)  =  y  <->  E. z  e.  A  ( 1st `  z )  =  y ) )
136, 12syl5rbb 249 . . 3  |-  ( Rel 
A  ->  ( E. z  e.  A  ( 1st `  z )  =  y  <->  y  e.  ran  ( x  e.  A  |->  ( 1st `  x
) ) ) )
141, 13bitrd 244 . 2  |-  ( Rel 
A  ->  ( y  e.  dom  A  <->  y  e.  ran  ( x  e.  A  |->  ( 1st `  x
) ) ) )
1514eqrdv 2281 1  |-  ( Rel 
A  ->  dom  A  =  ran  ( x  e.  A  |->  ( 1st `  x
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   E.wrex 2544    e. cmpt 4077   dom cdm 4689   ran crn 4690   Rel wrel 4694    Fn wfn 5250   ` cfv 5255   1stc1st 6120
This theorem is referenced by:  fidomdm  7138  dmct  23342
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-1st 6122  df-2nd 6123
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