Theorem mpt2xopxnop0 6940

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.

Step	Hyp	Ref	Expression
1		neq0 3795	. . 3 |- ( -. ( V F K ) = (/) <-> E. x x e. ( V F K ) )
2		mpt2xopn0yelv.f	. . . . . . 7 |- F = ( x e. _V , y e. ( 1st ` x ) |-> C )
3	2	dmmpt2ssx 6846	. . . . . 6 |- dom F C_ U_ x e. _V ( { x } X. ( 1st ` x ) )
4		elfvdm 5890	. . . . . . 7 |- ( x e. ( F ` <. V , K >. ) -> <. V , K >. e. dom F )
5		df-ov 6285	. . . . . . 7 |- ( V F K ) = ( F ` <. V , K >. )
6	4, 5	eleq2s 2575	. . . . . 6 |- ( x e. ( V F K ) -> <. V , K >. e. dom F )
7	3, 6	sseldi 3502	. . . . 5 |- ( x e. ( V F K ) -> <. V , K >. e. U_ x e. _V ( { x } X. ( 1st ` x ) ) )
8		fveq2 5864	. . . . . . 7 |- ( x = V -> ( 1st ` x ) = ( 1st ` V ) )
9	8	opeliunxp2 5139	. . . . . 6 |- ( <. V , K >. e. U_ x e. _V ( { x } X. ( 1st ` x ) ) <-> ( V e. _V /\ K e. ( 1st ` V ) ) )
10		eluni 4248	. . . . . . . . 9 |- ( K e. U. dom { V } <-> E. n ( K e. n /\ n e. dom { V } ) )
11		ne0i 3791	. . . . . . . . . . . . 13 |- ( n e. dom { V } -> dom { V } =/= (/) )
12	11	ad2antlr 726	. . . . . . . . . . . 12 |- ( ( ( K e. n /\ n e. dom { V } ) /\ V e. _V ) -> dom { V } =/= (/) )
13		dmsnn0 5471	. . . . . . . . . . . 12 |- ( V e. ( _V X. _V ) <-> dom { V } =/= (/) )
14	12, 13	sylibr 212	. . . . . . . . . . 11 |- ( ( ( K e. n /\ n e. dom { V } ) /\ V e. _V ) -> V e. ( _V X. _V ) )
15	14	ex 434	. . . . . . . . . 10 |- ( ( K e. n /\ n e. dom { V } ) -> ( V e. _V -> V e. ( _V X. _V ) ) )
16	15	exlimiv 1698	. . . . . . . . 9 |- ( E. n ( K e. n /\ n e. dom { V } ) -> ( V e. _V -> V e. ( _V X. _V ) ) )
17	10, 16	sylbi 195	. . . . . . . 8 |- ( K e. U. dom { V } -> ( V e. _V -> V e. ( _V X. _V ) ) )
18		1stval 6783	. . . . . . . 8 |- ( 1st ` V ) = U. dom { V }
19	17, 18	eleq2s 2575	. . . . . . 7 |- ( K e. ( 1st ` V ) -> ( V e. _V -> V e. ( _V X. _V ) ) )
20	19	impcom 430	. . . . . 6 |- ( ( V e. _V /\ K e. ( 1st ` V ) ) -> V e. ( _V X. _V ) )
21	9, 20	sylbi 195	. . . . 5 |- ( <. V , K >. e. U_ x e. _V ( { x } X. ( 1st ` x ) ) -> V e. ( _V X. _V ) )
22	7, 21	syl 16	. . . 4 |- ( x e. ( V F K ) -> V e. ( _V X. _V ) )
23	22	exlimiv 1698	. . 3 |- ( E. x x e. ( V F K ) -> V e. ( _V X. _V ) )
24	1, 23	sylbi 195	. 2 |- ( -. ( V F K ) = (/) -> V e. ( _V X. _V ) )
25	24	con1i 129	1 |- ( -. V e. ( _V X. _V ) -> ( V F K ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1379   E.wex 1596   e. wcel 1767   =/= wne 2662   _Vcvv 3113   (/)c0 3785   {csn 4027   <.cop 4033   U.cuni 4245   U_ciun 4325   X. cxp 4997   dom cdm 4999   ` cfv 5586  (class class class)co 6282   |-> cmpt2 6284   1stc1st 6779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782

This theorem is referenced by:  mpt2xopx0ov0  6941  mpt2xopxprcov0  6942