Theorem mpt2xopxnop0 6948

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 3751	. . 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 6851	. . . . . 6 |- dom F C_ U_ x e. _V ( { x } X. ( 1st ` x ) )
4		elfvdm 5877	. . . . . . 7 |- ( x e. ( F ` <. V , K >. ) -> <. V , K >. e. dom F )
5		df-ov 6283	. . . . . . 7 |- ( V F K ) = ( F ` <. V , K >. )
6	4, 5	eleq2s 2512	. . . . . 6 |- ( x e. ( V F K ) -> <. V , K >. e. dom F )
7	3, 6	sseldi 3442	. . . . 5 |- ( x e. ( V F K ) -> <. V , K >. e. U_ x e. _V ( { x } X. ( 1st ` x ) ) )
8		fveq2 5851	. . . . . . 7 |- ( x = V -> ( 1st ` x ) = ( 1st ` V ) )
9	8	opeliunxp2 4964	. . . . . 6 |- ( <. V , K >. e. U_ x e. _V ( { x } X. ( 1st ` x ) ) <-> ( V e. _V /\ K e. ( 1st ` V ) ) )
10		eluni 4196	. . . . . . . . 9 |- ( K e. U. dom { V } <-> E. n ( K e. n /\ n e. dom { V } ) )
11		ne0i 3746	. . . . . . . . . . . . 13 |- ( n e. dom { V } -> dom { V } =/= (/) )
12	11	ad2antlr 727	. . . . . . . . . . . 12 |- ( ( ( K e. n /\ n e. dom { V } ) /\ V e. _V ) -> dom { V } =/= (/) )
13		dmsnn0 5291	. . . . . . . . . . . 12 |- ( V e. ( _V X. _V ) <-> dom { V } =/= (/) )
14	12, 13	sylibr 214	. . . . . . . . . . 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 1745	. . . . . . . . 9 |- ( E. n ( K e. n /\ n e. dom { V } ) -> ( V e. _V -> V e. ( _V X. _V ) ) )
17	10, 16	sylbi 197	. . . . . . . 8 |- ( K e. U. dom { V } -> ( V e. _V -> V e. ( _V X. _V ) ) )
18		1stval 6788	. . . . . . . 8 |- ( 1st ` V ) = U. dom { V }
19	17, 18	eleq2s 2512	. . . . . . 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 197	. . . . 5 |- ( <. V , K >. e. U_ x e. _V ( { x } X. ( 1st ` x ) ) -> V e. ( _V X. _V ) )
22	7, 21	syl 17	. . . 4 |- ( x e. ( V F K ) -> V e. ( _V X. _V ) )
23	22	exlimiv 1745	. . 3 |- ( E. x x e. ( V F K ) -> V e. ( _V X. _V ) )
24	1, 23	sylbi 197	. 2 |- ( -. ( V F K ) = (/) -> V e. ( _V X. _V ) )
25	24	con1i 131	1 |- ( -. V e. ( _V X. _V ) -> ( V F K ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1407   E.wex 1635   e. wcel 1844   =/= wne 2600   _Vcvv 3061   (/)c0 3740   {csn 3974   <.cop 3980   U.cuni 4193   U_ciun 4273   X. cxp 4823   dom cdm 4825   ` cfv 5571  (class class class)co 6280   |-> cmpt2 6282   1stc1st 6784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576

This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-1st 6786  df-2nd 6787

This theorem is referenced by:  mpt2xopx0ov0  6949  mpt2xopxprcov0  6950