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Theorem mpt2xopn0yelv 6466
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 6418 . . . 4  |-  dom  F  C_ 
U_ x  e.  _V  ( { x }  X.  ( 1st `  x ) )
3 elfvdm 5759 . . . . 5  |-  ( N  e.  ( F `  <. <. V ,  W >. ,  K >. )  -> 
<. <. V ,  W >. ,  K >.  e.  dom  F )
4 df-ov 6086 . . . . 5  |-  ( <. V ,  W >. F K )  =  ( F `  <. <. V ,  W >. ,  K >. )
53, 4eleq2s 2530 . . . 4  |-  ( N  e.  ( <. V ,  W >. F K )  ->  <. <. V ,  W >. ,  K >.  e.  dom  F )
62, 5sseldi 3348 . . 3  |-  ( N  e.  ( <. V ,  W >. F K )  ->  <. <. V ,  W >. ,  K >.  e.  U_ x  e.  _V  ( { x }  X.  ( 1st `  x ) ) )
7 fveq2 5730 . . . . 5  |-  ( x  =  <. V ,  W >.  ->  ( 1st `  x
)  =  ( 1st `  <. V ,  W >. ) )
87opeliunxp2 5015 . . . 4  |-  ( <. <. V ,  W >. ,  K >.  e.  U_ x  e.  _V  ( { x }  X.  ( 1st `  x
) )  <->  ( <. V ,  W >.  e.  _V  /\  K  e.  ( 1st `  <. V ,  W >. ) ) )
98simprbi 452 . . 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 6361 . . 3  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( 1st `  <. V ,  W >. )  =  V )
1211eleq2d 2505 . 2  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( K  e.  ( 1st `  <. V ,  W >. )  <->  K  e.  V ) )
1310, 12syl5ib 212 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 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   {csn 3816   <.cop 3819   U_ciun 4095    X. cxp 4878   dom cdm 4880   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   1stc1st 6349
This theorem is referenced by:  mpt2xopynvov0g  6467  mpt2xopovel  6473  nbgrael  21440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352
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