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Theorem releldm2 6076
Description: Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)
Assertion
Ref Expression
releldm2  |-  ( Rel 
A  ->  ( B  e.  dom  A  <->  E. x  e.  A  ( 1st `  x )  =  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem releldm2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 elex 2692 . . 3  |-  ( B  e.  dom  A  ->  B  e.  _V )
21anim2i 339 . 2  |-  ( ( Rel  A  /\  B  e.  dom  A )  -> 
( Rel  A  /\  B  e.  _V )
)
3 id 19 . . . . 5  |-  ( ( 1st `  x )  =  B  ->  ( 1st `  x )  =  B )
4 vex 2684 . . . . . 6  |-  x  e. 
_V
5 1stexg 6058 . . . . . 6  |-  ( x  e.  _V  ->  ( 1st `  x )  e. 
_V )
64, 5ax-mp 5 . . . . 5  |-  ( 1st `  x )  e.  _V
73, 6eqeltrrdi 2229 . . . 4  |-  ( ( 1st `  x )  =  B  ->  B  e.  _V )
87rexlimivw 2543 . . 3  |-  ( E. x  e.  A  ( 1st `  x )  =  B  ->  B  e.  _V )
98anim2i 339 . 2  |-  ( ( Rel  A  /\  E. x  e.  A  ( 1st `  x )  =  B )  ->  ( Rel  A  /\  B  e. 
_V ) )
10 eldm2g 4730 . . . 4  |-  ( B  e.  _V  ->  ( B  e.  dom  A  <->  E. y <. B ,  y >.  e.  A ) )
1110adantl 275 . . 3  |-  ( ( Rel  A  /\  B  e.  _V )  ->  ( B  e.  dom  A  <->  E. y <. B ,  y >.  e.  A ) )
12 df-rel 4541 . . . . . . . . 9  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
13 ssel 3086 . . . . . . . . 9  |-  ( A 
C_  ( _V  X.  _V )  ->  ( x  e.  A  ->  x  e.  ( _V  X.  _V ) ) )
1412, 13sylbi 120 . . . . . . . 8  |-  ( Rel 
A  ->  ( x  e.  A  ->  x  e.  ( _V  X.  _V ) ) )
1514imp 123 . . . . . . 7  |-  ( ( Rel  A  /\  x  e.  A )  ->  x  e.  ( _V  X.  _V ) )
16 op1steq 6070 . . . . . . 7  |-  ( x  e.  ( _V  X.  _V )  ->  ( ( 1st `  x )  =  B  <->  E. y  x  =  <. B , 
y >. ) )
1715, 16syl 14 . . . . . 6  |-  ( ( Rel  A  /\  x  e.  A )  ->  (
( 1st `  x
)  =  B  <->  E. y  x  =  <. B , 
y >. ) )
1817rexbidva 2432 . . . . 5  |-  ( Rel 
A  ->  ( E. x  e.  A  ( 1st `  x )  =  B  <->  E. x  e.  A  E. y  x  =  <. B ,  y >.
) )
1918adantr 274 . . . 4  |-  ( ( Rel  A  /\  B  e.  _V )  ->  ( E. x  e.  A  ( 1st `  x )  =  B  <->  E. x  e.  A  E. y  x  =  <. B , 
y >. ) )
20 rexcom4 2704 . . . . 5  |-  ( E. x  e.  A  E. y  x  =  <. B ,  y >.  <->  E. y E. x  e.  A  x  =  <. B , 
y >. )
21 risset 2461 . . . . . 6  |-  ( <. B ,  y >.  e.  A  <->  E. x  e.  A  x  =  <. B , 
y >. )
2221exbii 1584 . . . . 5  |-  ( E. y <. B ,  y
>.  e.  A  <->  E. y E. x  e.  A  x  =  <. B , 
y >. )
2320, 22bitr4i 186 . . . 4  |-  ( E. x  e.  A  E. y  x  =  <. B ,  y >.  <->  E. y <. B ,  y >.  e.  A )
2419, 23syl6bb 195 . . 3  |-  ( ( Rel  A  /\  B  e.  _V )  ->  ( E. x  e.  A  ( 1st `  x )  =  B  <->  E. y <. B ,  y >.  e.  A ) )
2511, 24bitr4d 190 . 2  |-  ( ( Rel  A  /\  B  e.  _V )  ->  ( B  e.  dom  A  <->  E. x  e.  A  ( 1st `  x )  =  B ) )
262, 9, 25pm5.21nd 901 1  |-  ( Rel 
A  ->  ( B  e.  dom  A  <->  E. x  e.  A  ( 1st `  x )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331   E.wex 1468    e. wcel 1480   E.wrex 2415   _Vcvv 2681    C_ wss 3066   <.cop 3525    X. cxp 4532   dom cdm 4534   Rel wrel 4539   ` cfv 5118   1stc1st 6029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fo 5124  df-fv 5126  df-1st 6031  df-2nd 6032
This theorem is referenced by:  reldm  6077
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