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Theorem mpt2xopn0yelv 28194
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 6205 . . . 4  |-  dom  F  C_ 
U_ x  e.  _V  ( { x }  X.  ( 1st `  x ) )
3 elfvdm 5570 . . . . 5  |-  ( N  e.  ( F `  <. <. V ,  W >. ,  K >. )  -> 
<. <. V ,  W >. ,  K >.  e.  dom  F )
4 df-ov 5877 . . . . 5  |-  ( <. V ,  W >. F K )  =  ( F `  <. <. V ,  W >. ,  K >. )
53, 4eleq2s 2388 . . . 4  |-  ( N  e.  ( <. V ,  W >. F K )  ->  <. <. V ,  W >. ,  K >.  e.  dom  F )
62, 5sseldi 3191 . . 3  |-  ( N  e.  ( <. V ,  W >. F K )  ->  <. <. V ,  W >. ,  K >.  e.  U_ x  e.  _V  ( { x }  X.  ( 1st `  x ) ) )
7 fveq2 5541 . . . . 5  |-  ( x  =  <. V ,  W >.  ->  ( 1st `  x
)  =  ( 1st `  <. V ,  W >. ) )
87opeliunxp2 4840 . . . 4  |-  ( <. <. V ,  W >. ,  K >.  e.  U_ x  e.  _V  ( { x }  X.  ( 1st `  x
) )  <->  ( <. V ,  W >.  e.  _V  /\  K  e.  ( 1st `  <. V ,  W >. ) ) )
98simprbi 450 . . 3  |-  ( <. <. V ,  W >. ,  K >.  e.  U_ x  e.  _V  ( { x }  X.  ( 1st `  x
) )  ->  K  e.  ( 1st `  <. V ,  W >. )
)
106, 9syl 15 . 2  |-  ( N  e.  ( <. V ,  W >. F K )  ->  K  e.  ( 1st `  <. V ,  W >. ) )
11 op1stg 6148 . . 3  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( 1st `  <. V ,  W >. )  =  V )
1211eleq2d 2363 . 2  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( K  e.  ( 1st `  <. V ,  W >. )  <->  K  e.  V ) )
1310, 12syl5ib 210 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 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653   <.cop 3656   U_ciun 3921    X. cxp 4703   dom cdm 4705   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136
This theorem is referenced by:  mpt2xopynvov0g  28195  mpt2xopovel  28201
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139
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