ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  releldm2 Unicode version

Theorem releldm2 5937
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 2630 . . 3  |-  ( B  e.  dom  A  ->  B  e.  _V )
21anim2i 334 . 2  |-  ( ( Rel  A  /\  B  e.  dom  A )  -> 
( Rel  A  /\  B  e.  _V )
)
3 id 19 . . . . 5  |-  ( ( 1st `  x )  =  B  ->  ( 1st `  x )  =  B )
4 vex 2622 . . . . . 6  |-  x  e. 
_V
5 1stexg 5920 . . . . . 6  |-  ( x  e.  _V  ->  ( 1st `  x )  e. 
_V )
64, 5ax-mp 7 . . . . 5  |-  ( 1st `  x )  e.  _V
73, 6syl6eqelr 2179 . . . 4  |-  ( ( 1st `  x )  =  B  ->  B  e.  _V )
87rexlimivw 2485 . . 3  |-  ( E. x  e.  A  ( 1st `  x )  =  B  ->  B  e.  _V )
98anim2i 334 . 2  |-  ( ( Rel  A  /\  E. x  e.  A  ( 1st `  x )  =  B )  ->  ( Rel  A  /\  B  e. 
_V ) )
10 eldm2g 4620 . . . 4  |-  ( B  e.  _V  ->  ( B  e.  dom  A  <->  E. y <. B ,  y >.  e.  A ) )
1110adantl 271 . . 3  |-  ( ( Rel  A  /\  B  e.  _V )  ->  ( B  e.  dom  A  <->  E. y <. B ,  y >.  e.  A ) )
12 df-rel 4435 . . . . . . . . 9  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
13 ssel 3017 . . . . . . . . 9  |-  ( A 
C_  ( _V  X.  _V )  ->  ( x  e.  A  ->  x  e.  ( _V  X.  _V ) ) )
1412, 13sylbi 119 . . . . . . . 8  |-  ( Rel 
A  ->  ( x  e.  A  ->  x  e.  ( _V  X.  _V ) ) )
1514imp 122 . . . . . . 7  |-  ( ( Rel  A  /\  x  e.  A )  ->  x  e.  ( _V  X.  _V ) )
16 op1steq 5931 . . . . . . 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 2377 . . . . 5  |-  ( Rel 
A  ->  ( E. x  e.  A  ( 1st `  x )  =  B  <->  E. x  e.  A  E. y  x  =  <. B ,  y >.
) )
1918adantr 270 . . . 4  |-  ( ( Rel  A  /\  B  e.  _V )  ->  ( E. x  e.  A  ( 1st `  x )  =  B  <->  E. x  e.  A  E. y  x  =  <. B , 
y >. ) )
20 rexcom4 2642 . . . . 5  |-  ( E. x  e.  A  E. y  x  =  <. B ,  y >.  <->  E. y E. x  e.  A  x  =  <. B , 
y >. )
21 risset 2406 . . . . . 6  |-  ( <. B ,  y >.  e.  A  <->  E. x  e.  A  x  =  <. B , 
y >. )
2221exbii 1541 . . . . 5  |-  ( E. y <. B ,  y
>.  e.  A  <->  E. y E. x  e.  A  x  =  <. B , 
y >. )
2320, 22bitr4i 185 . . . 4  |-  ( E. x  e.  A  E. y  x  =  <. B ,  y >.  <->  E. y <. B ,  y >.  e.  A )
2419, 23syl6bb 194 . . 3  |-  ( ( Rel  A  /\  B  e.  _V )  ->  ( E. x  e.  A  ( 1st `  x )  =  B  <->  E. y <. B ,  y >.  e.  A ) )
2511, 24bitr4d 189 . 2  |-  ( ( Rel  A  /\  B  e.  _V )  ->  ( B  e.  dom  A  <->  E. x  e.  A  ( 1st `  x )  =  B ) )
262, 9, 25pm5.21nd 863 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 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   E.wrex 2360   _Vcvv 2619    C_ wss 2997   <.cop 3444    X. cxp 4426   dom cdm 4428   Rel wrel 4433   ` cfv 5002   1stc1st 5891
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fo 5008  df-fv 5010  df-1st 5893  df-2nd 5894
This theorem is referenced by:  reldm  5938
  Copyright terms: Public domain W3C validator