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Theorem releldm2 6170
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 2796 . . 3  |-  ( B  e.  dom  A  ->  B  e.  _V )
21anim2i 552 . 2  |-  ( ( Rel  A  /\  B  e.  dom  A )  -> 
( Rel  A  /\  B  e.  _V )
)
3 id 19 . . . . 5  |-  ( ( 1st `  x )  =  B  ->  ( 1st `  x )  =  B )
4 fvex 5539 . . . . 5  |-  ( 1st `  x )  e.  _V
53, 4syl6eqelr 2372 . . . 4  |-  ( ( 1st `  x )  =  B  ->  B  e.  _V )
65rexlimivw 2663 . . 3  |-  ( E. x  e.  A  ( 1st `  x )  =  B  ->  B  e.  _V )
76anim2i 552 . 2  |-  ( ( Rel  A  /\  E. x  e.  A  ( 1st `  x )  =  B )  ->  ( Rel  A  /\  B  e. 
_V ) )
8 eldm2g 4875 . . . 4  |-  ( B  e.  _V  ->  ( B  e.  dom  A  <->  E. y <. B ,  y >.  e.  A ) )
98adantl 452 . . 3  |-  ( ( Rel  A  /\  B  e.  _V )  ->  ( B  e.  dom  A  <->  E. y <. B ,  y >.  e.  A ) )
10 df-rel 4696 . . . . . . . . 9  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
11 ssel 3174 . . . . . . . . 9  |-  ( A 
C_  ( _V  X.  _V )  ->  ( x  e.  A  ->  x  e.  ( _V  X.  _V ) ) )
1210, 11sylbi 187 . . . . . . . 8  |-  ( Rel 
A  ->  ( x  e.  A  ->  x  e.  ( _V  X.  _V ) ) )
1312imp 418 . . . . . . 7  |-  ( ( Rel  A  /\  x  e.  A )  ->  x  e.  ( _V  X.  _V ) )
14 op1steq 6164 . . . . . . 7  |-  ( x  e.  ( _V  X.  _V )  ->  ( ( 1st `  x )  =  B  <->  E. y  x  =  <. B , 
y >. ) )
1513, 14syl 15 . . . . . 6  |-  ( ( Rel  A  /\  x  e.  A )  ->  (
( 1st `  x
)  =  B  <->  E. y  x  =  <. B , 
y >. ) )
1615rexbidva 2560 . . . . 5  |-  ( Rel 
A  ->  ( E. x  e.  A  ( 1st `  x )  =  B  <->  E. x  e.  A  E. y  x  =  <. B ,  y >.
) )
1716adantr 451 . . . 4  |-  ( ( Rel  A  /\  B  e.  _V )  ->  ( E. x  e.  A  ( 1st `  x )  =  B  <->  E. x  e.  A  E. y  x  =  <. B , 
y >. ) )
18 rexcom4 2807 . . . . 5  |-  ( E. x  e.  A  E. y  x  =  <. B ,  y >.  <->  E. y E. x  e.  A  x  =  <. B , 
y >. )
19 risset 2590 . . . . . 6  |-  ( <. B ,  y >.  e.  A  <->  E. x  e.  A  x  =  <. B , 
y >. )
2019exbii 1569 . . . . 5  |-  ( E. y <. B ,  y
>.  e.  A  <->  E. y E. x  e.  A  x  =  <. B , 
y >. )
2118, 20bitr4i 243 . . . 4  |-  ( E. x  e.  A  E. y  x  =  <. B ,  y >.  <->  E. y <. B ,  y >.  e.  A )
2217, 21syl6bb 252 . . 3  |-  ( ( Rel  A  /\  B  e.  _V )  ->  ( E. x  e.  A  ( 1st `  x )  =  B  <->  E. y <. B ,  y >.  e.  A ) )
239, 22bitr4d 247 . 2  |-  ( ( Rel  A  /\  B  e.  _V )  ->  ( B  e.  dom  A  <->  E. x  e.  A  ( 1st `  x )  =  B ) )
242, 7, 23pm5.21nd 868 1  |-  ( Rel 
A  ->  ( B  e.  dom  A  <->  E. x  e.  A  ( 1st `  x )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    C_ wss 3152   <.cop 3643    X. cxp 4687   dom cdm 4689   Rel wrel 4694   ` cfv 5255   1stc1st 6120
This theorem is referenced by:  reldm  6171
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-fv 5263  df-1st 6122  df-2nd 6123
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