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Theorem mpt2xopxnop0 6980
Description: If the first argument 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, is not an ordered pair, then the value of the operation is the empty set. (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
mpt2xopxnop0  |-  ( -.  V  e.  ( _V 
X.  _V )  ->  ( V F K )  =  (/) )
Distinct variable groups:    x, y    x, K    x, V    x, F
Allowed substitution hints:    C( x, y)    F( y)    K( y)    V( y)

Proof of Theorem mpt2xopxnop0
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 neq0 3733 . . 3  |-  ( -.  ( V F K )  =  (/)  <->  E. x  x  e.  ( V F K ) )
2 mpt2xopn0yelv.f . . . . . . 7  |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  C )
32dmmpt2ssx 6877 . . . . . 6  |-  dom  F  C_ 
U_ x  e.  _V  ( { x }  X.  ( 1st `  x ) )
4 elfvdm 5905 . . . . . . 7  |-  ( x  e.  ( F `  <. V ,  K >. )  ->  <. V ,  K >.  e.  dom  F )
5 df-ov 6311 . . . . . . 7  |-  ( V F K )  =  ( F `  <. V ,  K >. )
64, 5eleq2s 2567 . . . . . 6  |-  ( x  e.  ( V F K )  ->  <. V ,  K >.  e.  dom  F
)
73, 6sseldi 3416 . . . . 5  |-  ( x  e.  ( V F K )  ->  <. V ,  K >.  e.  U_ x  e.  _V  ( { x }  X.  ( 1st `  x
) ) )
8 fveq2 5879 . . . . . . 7  |-  ( x  =  V  ->  ( 1st `  x )  =  ( 1st `  V
) )
98opeliunxp2 4978 . . . . . 6  |-  ( <. V ,  K >.  e. 
U_ x  e.  _V  ( { x }  X.  ( 1st `  x ) )  <->  ( V  e. 
_V  /\  K  e.  ( 1st `  V ) ) )
10 eluni 4193 . . . . . . . . 9  |-  ( K  e.  U. dom  { V }  <->  E. n ( K  e.  n  /\  n  e.  dom  { V }
) )
11 ne0i 3728 . . . . . . . . . . . . 13  |-  ( n  e.  dom  { V }  ->  dom  { V }  =/=  (/) )
1211ad2antlr 741 . . . . . . . . . . . 12  |-  ( ( ( K  e.  n  /\  n  e.  dom  { V } )  /\  V  e.  _V )  ->  dom  { V }  =/=  (/) )
13 dmsnn0 5308 . . . . . . . . . . . 12  |-  ( V  e.  ( _V  X.  _V )  <->  dom  { V }  =/=  (/) )
1412, 13sylibr 217 . . . . . . . . . . 11  |-  ( ( ( K  e.  n  /\  n  e.  dom  { V } )  /\  V  e.  _V )  ->  V  e.  ( _V 
X.  _V ) )
1514ex 441 . . . . . . . . . 10  |-  ( ( K  e.  n  /\  n  e.  dom  { V } )  ->  ( V  e.  _V  ->  V  e.  ( _V  X.  _V ) ) )
1615exlimiv 1784 . . . . . . . . 9  |-  ( E. n ( K  e.  n  /\  n  e. 
dom  { V } )  ->  ( V  e. 
_V  ->  V  e.  ( _V  X.  _V )
) )
1710, 16sylbi 200 . . . . . . . 8  |-  ( K  e.  U. dom  { V }  ->  ( V  e.  _V  ->  V  e.  ( _V  X.  _V ) ) )
18 1stval 6814 . . . . . . . 8  |-  ( 1st `  V )  =  U. dom  { V }
1917, 18eleq2s 2567 . . . . . . 7  |-  ( K  e.  ( 1st `  V
)  ->  ( V  e.  _V  ->  V  e.  ( _V  X.  _V )
) )
2019impcom 437 . . . . . 6  |-  ( ( V  e.  _V  /\  K  e.  ( 1st `  V ) )  ->  V  e.  ( _V  X.  _V ) )
219, 20sylbi 200 . . . . 5  |-  ( <. V ,  K >.  e. 
U_ x  e.  _V  ( { x }  X.  ( 1st `  x ) )  ->  V  e.  ( _V  X.  _V )
)
227, 21syl 17 . . . 4  |-  ( x  e.  ( V F K )  ->  V  e.  ( _V  X.  _V ) )
2322exlimiv 1784 . . 3  |-  ( E. x  x  e.  ( V F K )  ->  V  e.  ( _V  X.  _V )
)
241, 23sylbi 200 . 2  |-  ( -.  ( V F K )  =  (/)  ->  V  e.  ( _V  X.  _V ) )
2524con1i 134 1  |-  ( -.  V  e.  ( _V 
X.  _V )  ->  ( V F K )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904    =/= wne 2641   _Vcvv 3031   (/)c0 3722   {csn 3959   <.cop 3965   U.cuni 4190   U_ciun 4269    X. cxp 4837   dom cdm 4839   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   1stc1st 6810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813
This theorem is referenced by:  mpt2xopx0ov0  6981  mpt2xopxprcov0  6982
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