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Theorem mpt2xopn0yelv 6393
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 6348 . . . 4  |-  dom  F  C_ 
U_ x  e.  _V  ( { x }  X.  ( 1st `  x ) )
3 elfvdm 5690 . . . . 5  |-  ( N  e.  ( F `  <. <. V ,  W >. ,  K >. )  -> 
<. <. V ,  W >. ,  K >.  e.  dom  F )
4 df-ov 6016 . . . . 5  |-  ( <. V ,  W >. F K )  =  ( F `  <. <. V ,  W >. ,  K >. )
53, 4eleq2s 2472 . . . 4  |-  ( N  e.  ( <. V ,  W >. F K )  ->  <. <. V ,  W >. ,  K >.  e.  dom  F )
62, 5sseldi 3282 . . 3  |-  ( N  e.  ( <. V ,  W >. F K )  ->  <. <. V ,  W >. ,  K >.  e.  U_ x  e.  _V  ( { x }  X.  ( 1st `  x ) ) )
7 fveq2 5661 . . . . 5  |-  ( x  =  <. V ,  W >.  ->  ( 1st `  x
)  =  ( 1st `  <. V ,  W >. ) )
87opeliunxp2 4946 . . . 4  |-  ( <. <. V ,  W >. ,  K >.  e.  U_ x  e.  _V  ( { x }  X.  ( 1st `  x
) )  <->  ( <. V ,  W >.  e.  _V  /\  K  e.  ( 1st `  <. V ,  W >. ) ) )
98simprbi 451 . . 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 6291 . . 3  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( 1st `  <. V ,  W >. )  =  V )
1211eleq2d 2447 . 2  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( K  e.  ( 1st `  <. V ,  W >. )  <->  K  e.  V ) )
1310, 12syl5ib 211 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 359    = wceq 1649    e. wcel 1717   _Vcvv 2892   {csn 3750   <.cop 3753   U_ciun 4028    X. cxp 4809   dom cdm 4811   ` cfv 5387  (class class class)co 6013    e. cmpt2 6015   1stc1st 6279
This theorem is referenced by:  mpt2xopynvov0g  6394  mpt2xopovel  6400  nbgrael  21297
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282
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