Theorem mpoxopn0yelv 6325

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 6285	. . . 4 |- dom F C_ U_ x e. _V ( { x } X. ( 1st ` x ) )
3	1	mpofun 6047	. . . . . . 7 |- Fun F
4		funrel 5288	. . . . . . 7 |- ( Fun F -> Rel F )
5	3, 4	ax-mp 5	. . . . . 6 |- Rel F
6		relelfvdm 5608	. . . . . 6 |- ( ( Rel F /\ N e. ( F ` <. <. V , W >. , K >. ) ) -> <. <. V , W >. , K >. e. dom F )
7	5, 6	mpan 424	. . . . 5 |- ( N e. ( F ` <. <. V , W >. , K >. ) -> <. <. V , W >. , K >. e. dom F )
8		df-ov 5947	. . . . 5 |- ( <. V , W >. F K ) = ( F ` <. <. V , W >. , K >. )
9	7, 8	eleq2s 2300	. . . 4 |- ( N e. ( <. V , W >. F K ) -> <. <. V , W >. , K >. e. dom F )
10	2, 9	sselid 3191	. . 3 |- ( N e. ( <. V , W >. F K ) -> <. <. V , W >. , K >. e. U_ x e. _V ( { x } X. ( 1st ` x ) ) )
11		fveq2 5576	. . . . 5 |- ( x = <. V , W >. -> ( 1st ` x ) = ( 1st ` <. V , W >. ) )
12	11	opeliunxp2 4818	. . . 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 275	. . 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 6236	. . 3 |- ( ( V e. X /\ W e. Y ) -> ( 1st ` <. V , W >. ) = V )
16	15	eleq2d 2275	. 2 |- ( ( V e. X /\ W e. Y ) -> ( K e. ( 1st ` <. V , W >. ) <-> K e. V ) )
17	14, 16	imbitrid 154	1 |- ( ( V e. X /\ W e. Y ) -> ( N e. ( <. V , W >. F K ) -> K e. V ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   _Vcvv 2772   {csn 3633   <.cop 3636   U_ciun 3927   X. cxp 4673   dom cdm 4675   Rel wrel 4680   Fun wfun 5265   ` cfv 5271  (class class class)co 5944   e. cmpo 5946   1stc1st 6224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227

This theorem is referenced by:  mpoxopovel  6327