Theorem mpoxopn0yelv 6198

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
mpoxopn0yelv.f	|- F = ( x e. _V , y e. ( 1st ` x ) |-> C )

Assertion

Ref	Expression
mpoxopn0yelv	|- ( ( 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 mpoxopn0yelv

Step	Hyp	Ref	Expression
1		mpoxopn0yelv.f	. . . . 5 |- F = ( x e. _V , y e. ( 1st ` x ) |-> C )
2	1	dmmpossx 6159	. . . 4 |- dom F C_ U_ x e. _V ( { x } X. ( 1st ` x ) )
3	1	mpofun 5935	. . . . . . 7 |- Fun F
4		funrel 5199	. . . . . . 7 |- ( Fun F -> Rel F )
5	3, 4	ax-mp 5	. . . . . 6 |- Rel F
6		relelfvdm 5512	. . . . . 6 |- ( ( Rel F /\ N e. ( F ` <. <. V , W >. , K >. ) ) -> <. <. V , W >. , K >. e. dom F )
7	5, 6	mpan 421	. . . . 5 |- ( N e. ( F ` <. <. V , W >. , K >. ) -> <. <. V , W >. , K >. e. dom F )
8		df-ov 5839	. . . . 5 |- ( <. V , W >. F K ) = ( F ` <. <. V , W >. , K >. )
9	7, 8	eleq2s 2259	. . . 4 |- ( N e. ( <. V , W >. F K ) -> <. <. V , W >. , K >. e. dom F )
10	2, 9	sseldi 3135	. . 3 |- ( N e. ( <. V , W >. F K ) -> <. <. V , W >. , K >. e. U_ x e. _V ( { x } X. ( 1st ` x ) ) )
11		fveq2 5480	. . . . 5 |- ( x = <. V , W >. -> ( 1st ` x ) = ( 1st ` <. V , W >. ) )
12	11	opeliunxp2 4738	. . . 4 |- ( <. <. V , W >. , K >. e. U_ x e. _V ( { x } X. ( 1st ` x ) ) <-> ( <. V , W >. e. _V /\ K e. ( 1st ` <. V , W >. ) ) )
13	12	simprbi 273	. . 3 |- ( <. <. V , W >. , K >. e. U_ x e. _V ( { x } X. ( 1st ` x ) ) -> K e. ( 1st ` <. V , W >. ) )
14	10, 13	syl 14	. 2 |- ( N e. ( <. V , W >. F K ) -> K e. ( 1st ` <. V , W >. ) )
15		op1stg 6110	. . 3 |- ( ( V e. X /\ W e. Y ) -> ( 1st ` <. V , W >. ) = V )
16	15	eleq2d 2234	. 2 |- ( ( V e. X /\ W e. Y ) -> ( K e. ( 1st ` <. V , W >. ) <-> K e. V ) )
17	14, 16	syl5ib 153	1 |- ( ( V e. X /\ W e. Y ) -> ( N e. ( <. V , W >. F K ) -> K e. V ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1342   e. wcel 2135   _Vcvv 2721   {csn 3570   <.cop 3573   U_ciun 3860   X. cxp 4596   dom cdm 4598   Rel wrel 4603   Fun wfun 5176   ` cfv 5182  (class class class)co 5836   e. cmpo 5838   1stc1st 6098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fv 5190  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101

This theorem is referenced by:  mpoxopovel  6200