Theorem mpoxopn0yelv 6348

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 6308	. . . 4 |- dom F C_ U_ x e. _V ( { x } X. ( 1st ` x ) )
3	1	mpofun 6070	. . . . . . 7 |- Fun F
4		funrel 5307	. . . . . . 7 |- ( Fun F -> Rel F )
5	3, 4	ax-mp 5	. . . . . 6 |- Rel F
6		relelfvdm 5631	. . . . . 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 5970	. . . . 5 |- ( <. V , W >. F K ) = ( F ` <. <. V , W >. , K >. )
9	7, 8	eleq2s 2302	. . . 4 |- ( N e. ( <. V , W >. F K ) -> <. <. V , W >. , K >. e. dom F )
10	2, 9	sselid 3199	. . 3 |- ( N e. ( <. V , W >. F K ) -> <. <. V , W >. , K >. e. U_ x e. _V ( { x } X. ( 1st ` x ) ) )
11		fveq2 5599	. . . . 5 |- ( x = <. V , W >. -> ( 1st ` x ) = ( 1st ` <. V , W >. ) )
12	11	opeliunxp2 4836	. . . 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 6259	. . 3 |- ( ( V e. X /\ W e. Y ) -> ( 1st ` <. V , W >. ) = V )
16	15	eleq2d 2277	. 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 2178   _Vcvv 2776   {csn 3643   <.cop 3646   U_ciun 3941   X. cxp 4691   dom cdm 4693   Rel wrel 4698   Fun wfun 5284   ` cfv 5290  (class class class)co 5967   e. cmpo 5969   1stc1st 6247

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250

This theorem is referenced by:  mpoxopovel  6350