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Theorem reldm 6399
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 6398 . . 3  |-  ( Rel 
A  ->  ( y  e.  dom  A  <->  E. z  e.  A  ( 1st `  z )  =  y ) )
2 fvex 5743 . . . . . 6  |-  ( 1st `  x )  e.  _V
3 eqid 2437 . . . . . 6  |-  ( x  e.  A  |->  ( 1st `  x ) )  =  ( x  e.  A  |->  ( 1st `  x
) )
42, 3fnmpti 5574 . . . . 5  |-  ( x  e.  A  |->  ( 1st `  x ) )  Fn  A
5 fvelrnb 5775 . . . . 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 5729 . . . . . . . 8  |-  ( x  =  z  ->  ( 1st `  x )  =  ( 1st `  z
) )
8 fvex 5743 . . . . . . . 8  |-  ( 1st `  z )  e.  _V
97, 3, 8fvmpt 5807 . . . . . . 7  |-  ( z  e.  A  ->  (
( x  e.  A  |->  ( 1st `  x
) ) `  z
)  =  ( 1st `  z ) )
109eqeq1d 2445 . . . . . 6  |-  ( z  e.  A  ->  (
( ( x  e.  A  |->  ( 1st `  x
) ) `  z
)  =  y  <->  ( 1st `  z )  =  y ) )
1110rexbiia 2739 . . . . 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 251 . . 3  |-  ( Rel 
A  ->  ( E. z  e.  A  ( 1st `  z )  =  y  <->  y  e.  ran  ( x  e.  A  |->  ( 1st `  x
) ) ) )
141, 13bitrd 246 . 2  |-  ( Rel 
A  ->  ( y  e.  dom  A  <->  y  e.  ran  ( x  e.  A  |->  ( 1st `  x
) ) ) )
1514eqrdv 2435 1  |-  ( Rel 
A  ->  dom  A  =  ran  ( x  e.  A  |->  ( 1st `  x
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   E.wrex 2707    e. cmpt 4267   dom cdm 4879   ran crn 4880   Rel wrel 4884    Fn wfn 5450   ` cfv 5455   1stc1st 6348
This theorem is referenced by:  fidomdm  7389  dmct  24107
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-iota 5419  df-fun 5457  df-fn 5458  df-fv 5463  df-1st 6350  df-2nd 6351
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