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Theorem releldm2 5839
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 2583 . . 3  |-  ( B  e.  dom  A  ->  B  e.  _V )
21anim2i 328 . 2  |-  ( ( Rel  A  /\  B  e.  dom  A )  -> 
( Rel  A  /\  B  e.  _V )
)
3 id 19 . . . . 5  |-  ( ( 1st `  x )  =  B  ->  ( 1st `  x )  =  B )
4 vex 2577 . . . . . 6  |-  x  e. 
_V
5 1stexg 5822 . . . . . 6  |-  ( x  e.  _V  ->  ( 1st `  x )  e. 
_V )
64, 5ax-mp 7 . . . . 5  |-  ( 1st `  x )  e.  _V
73, 6syl6eqelr 2145 . . . 4  |-  ( ( 1st `  x )  =  B  ->  B  e.  _V )
87rexlimivw 2446 . . 3  |-  ( E. x  e.  A  ( 1st `  x )  =  B  ->  B  e.  _V )
98anim2i 328 . 2  |-  ( ( Rel  A  /\  E. x  e.  A  ( 1st `  x )  =  B )  ->  ( Rel  A  /\  B  e. 
_V ) )
10 eldm2g 4559 . . . 4  |-  ( B  e.  _V  ->  ( B  e.  dom  A  <->  E. y <. B ,  y >.  e.  A ) )
1110adantl 266 . . 3  |-  ( ( Rel  A  /\  B  e.  _V )  ->  ( B  e.  dom  A  <->  E. y <. B ,  y >.  e.  A ) )
12 df-rel 4380 . . . . . . . . 9  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
13 ssel 2967 . . . . . . . . 9  |-  ( A 
C_  ( _V  X.  _V )  ->  ( x  e.  A  ->  x  e.  ( _V  X.  _V ) ) )
1412, 13sylbi 118 . . . . . . . 8  |-  ( Rel 
A  ->  ( x  e.  A  ->  x  e.  ( _V  X.  _V ) ) )
1514imp 119 . . . . . . 7  |-  ( ( Rel  A  /\  x  e.  A )  ->  x  e.  ( _V  X.  _V ) )
16 op1steq 5833 . . . . . . 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 2340 . . . . 5  |-  ( Rel 
A  ->  ( E. x  e.  A  ( 1st `  x )  =  B  <->  E. x  e.  A  E. y  x  =  <. B ,  y >.
) )
1918adantr 265 . . . 4  |-  ( ( Rel  A  /\  B  e.  _V )  ->  ( E. x  e.  A  ( 1st `  x )  =  B  <->  E. x  e.  A  E. y  x  =  <. B , 
y >. ) )
20 rexcom4 2594 . . . . 5  |-  ( E. x  e.  A  E. y  x  =  <. B ,  y >.  <->  E. y E. x  e.  A  x  =  <. B , 
y >. )
21 risset 2369 . . . . . 6  |-  ( <. B ,  y >.  e.  A  <->  E. x  e.  A  x  =  <. B , 
y >. )
2221exbii 1512 . . . . 5  |-  ( E. y <. B ,  y
>.  e.  A  <->  E. y E. x  e.  A  x  =  <. B , 
y >. )
2320, 22bitr4i 180 . . . 4  |-  ( E. x  e.  A  E. y  x  =  <. B ,  y >.  <->  E. y <. B ,  y >.  e.  A )
2419, 23syl6bb 189 . . 3  |-  ( ( Rel  A  /\  B  e.  _V )  ->  ( E. x  e.  A  ( 1st `  x )  =  B  <->  E. y <. B ,  y >.  e.  A ) )
2511, 24bitr4d 184 . 2  |-  ( ( Rel  A  /\  B  e.  _V )  ->  ( B  e.  dom  A  <->  E. x  e.  A  ( 1st `  x )  =  B ) )
262, 9, 25pm5.21nd 836 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 101    <-> wb 102    = wceq 1259   E.wex 1397    e. wcel 1409   E.wrex 2324   _Vcvv 2574    C_ wss 2945   <.cop 3406    X. cxp 4371   dom cdm 4373   Rel wrel 4378   ` 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:  reldm  5840
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