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Theorem mpoxopn0yelv 6139
Description: If there is an element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, then the second argument is an element of the first component of the first argument. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
Hypothesis
Ref Expression
mpoxopn0yelv.f  |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  C )
Assertion
Ref Expression
mpoxopn0yelv  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( N  e.  (
<. V ,  W >. F K )  ->  K  e.  V ) )
Distinct variable groups:    x, y    x, K    x, V    x, W
Allowed substitution hints:    C( x, y)    F( x, y)    K( y)    N( x, y)    V( y)    W( y)    X( x, y)    Y( x, y)

Proof of Theorem mpoxopn0yelv
StepHypRef Expression
1 mpoxopn0yelv.f . . . . 5  |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  C )
21dmmpossx 6100 . . . 4  |-  dom  F  C_ 
U_ x  e.  _V  ( { x }  X.  ( 1st `  x ) )
31mpofun 5876 . . . . . . 7  |-  Fun  F
4 funrel 5143 . . . . . . 7  |-  ( Fun 
F  ->  Rel  F )
53, 4ax-mp 5 . . . . . 6  |-  Rel  F
6 relelfvdm 5456 . . . . . 6  |-  ( ( Rel  F  /\  N  e.  ( F `  <. <. V ,  W >. ,  K >. ) )  ->  <. <. V ,  W >. ,  K >.  e.  dom  F )
75, 6mpan 420 . . . . 5  |-  ( N  e.  ( F `  <. <. V ,  W >. ,  K >. )  -> 
<. <. V ,  W >. ,  K >.  e.  dom  F )
8 df-ov 5780 . . . . 5  |-  ( <. V ,  W >. F K )  =  ( F `  <. <. V ,  W >. ,  K >. )
97, 8eleq2s 2234 . . . 4  |-  ( N  e.  ( <. V ,  W >. F K )  ->  <. <. V ,  W >. ,  K >.  e.  dom  F )
102, 9sseldi 3095 . . 3  |-  ( N  e.  ( <. V ,  W >. F K )  ->  <. <. V ,  W >. ,  K >.  e.  U_ x  e.  _V  ( { x }  X.  ( 1st `  x ) ) )
11 fveq2 5424 . . . . 5  |-  ( x  =  <. V ,  W >.  ->  ( 1st `  x
)  =  ( 1st `  <. V ,  W >. ) )
1211opeliunxp2 4682 . . . 4  |-  ( <. <. V ,  W >. ,  K >.  e.  U_ x  e.  _V  ( { x }  X.  ( 1st `  x
) )  <->  ( <. V ,  W >.  e.  _V  /\  K  e.  ( 1st `  <. V ,  W >. ) ) )
1312simprbi 273 . . 3  |-  ( <. <. V ,  W >. ,  K >.  e.  U_ x  e.  _V  ( { x }  X.  ( 1st `  x
) )  ->  K  e.  ( 1st `  <. V ,  W >. )
)
1410, 13syl 14 . 2  |-  ( N  e.  ( <. V ,  W >. F K )  ->  K  e.  ( 1st `  <. V ,  W >. ) )
15 op1stg 6051 . . 3  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( 1st `  <. V ,  W >. )  =  V )
1615eleq2d 2209 . 2  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( K  e.  ( 1st `  <. V ,  W >. )  <->  K  e.  V ) )
1714, 16syl5ib 153 1  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( N  e.  (
<. V ,  W >. F K )  ->  K  e.  V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   _Vcvv 2686   {csn 3527   <.cop 3530   U_ciun 3816    X. cxp 4540   dom cdm 4542   Rel wrel 4547   Fun wfun 5120   ` cfv 5126  (class class class)co 5777    e. cmpo 5779   1stc1st 6039
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 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134  ax-un 4358
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-iun 3818  df-br 3933  df-opab 3993  df-mpt 3994  df-id 4218  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555  df-iota 5091  df-fun 5128  df-fv 5134  df-ov 5780  df-oprab 5781  df-mpo 5782  df-1st 6041  df-2nd 6042
This theorem is referenced by:  mpoxopovel  6141
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