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Theorem mpt2xopn0yelv 6959
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
mpt2xopn0yelv.f  |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  C )
Assertion
Ref Expression
mpt2xopn0yelv  |-  ( ( 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 mpt2xopn0yelv
StepHypRef Expression
1 mpt2xopn0yelv.f . . . . 5  |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  C )
21dmmpt2ssx 6864 . . . 4  |-  dom  F  C_ 
U_ x  e.  _V  ( { x }  X.  ( 1st `  x ) )
3 elfvdm 5898 . . . . 5  |-  ( N  e.  ( F `  <. <. V ,  W >. ,  K >. )  -> 
<. <. V ,  W >. ,  K >.  e.  dom  F )
4 df-ov 6299 . . . . 5  |-  ( <. V ,  W >. F K )  =  ( F `  <. <. V ,  W >. ,  K >. )
53, 4eleq2s 2565 . . . 4  |-  ( N  e.  ( <. V ,  W >. F K )  ->  <. <. V ,  W >. ,  K >.  e.  dom  F )
62, 5sseldi 3497 . . 3  |-  ( N  e.  ( <. V ,  W >. F K )  ->  <. <. V ,  W >. ,  K >.  e.  U_ x  e.  _V  ( { x }  X.  ( 1st `  x ) ) )
7 fveq2 5872 . . . . 5  |-  ( x  =  <. V ,  W >.  ->  ( 1st `  x
)  =  ( 1st `  <. V ,  W >. ) )
87opeliunxp2 5151 . . . 4  |-  ( <. <. V ,  W >. ,  K >.  e.  U_ x  e.  _V  ( { x }  X.  ( 1st `  x
) )  <->  ( <. V ,  W >.  e.  _V  /\  K  e.  ( 1st `  <. V ,  W >. ) ) )
98simprbi 464 . . 3  |-  ( <. <. V ,  W >. ,  K >.  e.  U_ x  e.  _V  ( { x }  X.  ( 1st `  x
) )  ->  K  e.  ( 1st `  <. V ,  W >. )
)
106, 9syl 16 . 2  |-  ( N  e.  ( <. V ,  W >. F K )  ->  K  e.  ( 1st `  <. V ,  W >. ) )
11 op1stg 6811 . . 3  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( 1st `  <. V ,  W >. )  =  V )
1211eleq2d 2527 . 2  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( K  e.  ( 1st `  <. V ,  W >. )  <->  K  e.  V ) )
1310, 12syl5ib 219 1  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( N  e.  (
<. V ,  W >. F K )  ->  K  e.  V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109   {csn 4032   <.cop 4038   U_ciun 4332    X. cxp 5006   dom cdm 5008   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   1stc1st 6797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800
This theorem is referenced by:  mpt2xopynvov0g  6960  mpt2xopovel  6966  nbgrael  24553
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