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Theorem releldm2 6389
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 2956 . . 3  |-  ( B  e.  dom  A  ->  B  e.  _V )
21anim2i 553 . 2  |-  ( ( Rel  A  /\  B  e.  dom  A )  -> 
( Rel  A  /\  B  e.  _V )
)
3 id 20 . . . . 5  |-  ( ( 1st `  x )  =  B  ->  ( 1st `  x )  =  B )
4 fvex 5734 . . . . 5  |-  ( 1st `  x )  e.  _V
53, 4syl6eqelr 2524 . . . 4  |-  ( ( 1st `  x )  =  B  ->  B  e.  _V )
65rexlimivw 2818 . . 3  |-  ( E. x  e.  A  ( 1st `  x )  =  B  ->  B  e.  _V )
76anim2i 553 . 2  |-  ( ( Rel  A  /\  E. x  e.  A  ( 1st `  x )  =  B )  ->  ( Rel  A  /\  B  e. 
_V ) )
8 eldm2g 5058 . . . 4  |-  ( B  e.  _V  ->  ( B  e.  dom  A  <->  E. y <. B ,  y >.  e.  A ) )
98adantl 453 . . 3  |-  ( ( Rel  A  /\  B  e.  _V )  ->  ( B  e.  dom  A  <->  E. y <. B ,  y >.  e.  A ) )
10 df-rel 4877 . . . . . . . . 9  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
11 ssel 3334 . . . . . . . . 9  |-  ( A 
C_  ( _V  X.  _V )  ->  ( x  e.  A  ->  x  e.  ( _V  X.  _V ) ) )
1210, 11sylbi 188 . . . . . . . 8  |-  ( Rel 
A  ->  ( x  e.  A  ->  x  e.  ( _V  X.  _V ) ) )
1312imp 419 . . . . . . 7  |-  ( ( Rel  A  /\  x  e.  A )  ->  x  e.  ( _V  X.  _V ) )
14 op1steq 6383 . . . . . . 7  |-  ( x  e.  ( _V  X.  _V )  ->  ( ( 1st `  x )  =  B  <->  E. y  x  =  <. B , 
y >. ) )
1513, 14syl 16 . . . . . 6  |-  ( ( Rel  A  /\  x  e.  A )  ->  (
( 1st `  x
)  =  B  <->  E. y  x  =  <. B , 
y >. ) )
1615rexbidva 2714 . . . . 5  |-  ( Rel 
A  ->  ( E. x  e.  A  ( 1st `  x )  =  B  <->  E. x  e.  A  E. y  x  =  <. B ,  y >.
) )
1716adantr 452 . . . 4  |-  ( ( Rel  A  /\  B  e.  _V )  ->  ( E. x  e.  A  ( 1st `  x )  =  B  <->  E. x  e.  A  E. y  x  =  <. B , 
y >. ) )
18 rexcom4 2967 . . . . 5  |-  ( E. x  e.  A  E. y  x  =  <. B ,  y >.  <->  E. y E. x  e.  A  x  =  <. B , 
y >. )
19 risset 2745 . . . . . 6  |-  ( <. B ,  y >.  e.  A  <->  E. x  e.  A  x  =  <. B , 
y >. )
2019exbii 1592 . . . . 5  |-  ( E. y <. B ,  y
>.  e.  A  <->  E. y E. x  e.  A  x  =  <. B , 
y >. )
2118, 20bitr4i 244 . . . 4  |-  ( E. x  e.  A  E. y  x  =  <. B ,  y >.  <->  E. y <. B ,  y >.  e.  A )
2217, 21syl6bb 253 . . 3  |-  ( ( Rel  A  /\  B  e.  _V )  ->  ( E. x  e.  A  ( 1st `  x )  =  B  <->  E. y <. B ,  y >.  e.  A ) )
239, 22bitr4d 248 . 2  |-  ( ( Rel  A  /\  B  e.  _V )  ->  ( B  e.  dom  A  <->  E. x  e.  A  ( 1st `  x )  =  B ) )
242, 7, 23pm5.21nd 869 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 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   E.wrex 2698   _Vcvv 2948    C_ wss 3312   <.cop 3809    X. cxp 4868   dom cdm 4870   Rel wrel 4875   ` cfv 5446   1stc1st 6339
This theorem is referenced by:  reldm  6390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-1st 6341  df-2nd 6342
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