Theorem mpt2xopn0yelv 5877

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
mpt2xopn0yelv.f	|- F = ( x e. _V , y e. ( 1st ` x ) |-> C )

Assertion

Ref	Expression
mpt2xopn0yelv	|- ( ( 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 mpt2xopn0yelv

Step	Hyp	Ref	Expression
1		mpt2xopn0yelv.f	. . . . 5 |- F = ( x e. _V , y e. ( 1st ` x ) |-> C )
2	1	dmmpt2ssx 5845	. . . 4 |- dom F C_ U_ x e. _V ( { x } X. ( 1st ` x ) )
3	1	mpt2fun 5623	. . . . . . 7 |- Fun F
4		funrel 4939	. . . . . . 7 |- ( Fun F -> Rel F )
5	3, 4	ax-mp 7	. . . . . 6 |- Rel F
6		relelfvdm 5226	. . . . . 6 |- ( ( Rel F /\ N e. ( F ` <. <. V , W >. , K >. ) ) -> <. <. V , W >. , K >. e. dom F )
7	5, 6	mpan 414	. . . . 5 |- ( N e. ( F ` <. <. V , W >. , K >. ) -> <. <. V , W >. , K >. e. dom F )
8		df-ov 5535	. . . . 5 |- ( <. V , W >. F K ) = ( F ` <. <. V , W >. , K >. )
9	7, 8	eleq2s 2173	. . . 4 |- ( N e. ( <. V , W >. F K ) -> <. <. V , W >. , K >. e. dom F )
10	2, 9	sseldi 2997	. . 3 |- ( N e. ( <. V , W >. F K ) -> <. <. V , W >. , K >. e. U_ x e. _V ( { x } X. ( 1st ` x ) ) )
11		fveq2 5198	. . . . 5 |- ( x = <. V , W >. -> ( 1st ` x ) = ( 1st ` <. V , W >. ) )
12	11	opeliunxp2 4494	. . . 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 269	. . 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 5797	. . 3 |- ( ( V e. X /\ W e. Y ) -> ( 1st ` <. V , W >. ) = V )
16	15	eleq2d 2148	. 2 |- ( ( V e. X /\ W e. Y ) -> ( K e. ( 1st ` <. V , W >. ) <-> K e. V ) )
17	14, 16	syl5ib 152	1 |- ( ( V e. X /\ W e. Y ) -> ( N e. ( <. V , W >. F K ) -> K e. V ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   _Vcvv 2601   {csn 3398   <.cop 3401   U_ciun 3678   X. cxp 4361   dom cdm 4363   Rel wrel 4368   Fun wfun 4916   ` cfv 4922  (class class class)co 5532   |-> cmpt2 5534   1stc1st 5785

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788

This theorem is referenced by:  mpt2xopovel  5879