Theorem mpt2xopxprcov0 6982

Description: If the components of 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, are not sets, 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
mpt2xopxprcov0	|- ( -. ( V e. _V /\ W e. _V ) -> ( <. V , W >. F K ) = (/) )

Distinct variable groups:   x, y   x, K   x, V   x, W   x, F

Allowed substitution hints:   C( x, y)   F( y)   K( y)   V( y)   W( y)

Proof of Theorem mpt2xopxprcov0

Step	Hyp	Ref	Expression
1		opelxp 4869	. 2 |- ( <. V , W >. e. ( _V X. _V ) <-> ( V e. _V /\ W e. _V ) )
2		mpt2xopn0yelv.f	. . 3 |- F = ( x e. _V , y e. ( 1st ` x ) |-> C )
3	2	mpt2xopxnop0 6980	. 2 |- ( -. <. V , W >. e. ( _V X. _V ) -> ( <. V , W >. F K ) = (/) )
4	1, 3	sylnbir 314	1 |- ( -. ( V e. _V /\ W e. _V ) -> ( <. V , W >. F K ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 376   = wceq 1452   e. wcel 1904   _Vcvv 3031   (/)c0 3722   <.cop 3965   X. cxp 4837   ` 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:  mpt2xopynvov0  6983