Theorem mpt2xopn0yelv 7339

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 7235	. . . 4 |- dom F C_ U_ x e. _V ( { x } X. ( 1st ` x ) )
3		elfvdm 6220	. . . . 5 |- ( N e. ( F ` <. <. V , W >. , K >. ) -> <. <. V , W >. , K >. e. dom F )
4		df-ov 6653	. . . . 5 |- ( <. V , W >. F K ) = ( F ` <. <. V , W >. , K >. )
5	3, 4	eleq2s 2719	. . . 4 |- ( N e. ( <. V , W >. F K ) -> <. <. V , W >. , K >. e. dom F )
6	2, 5	sseldi 3601	. . 3 |- ( N e. ( <. V , W >. F K ) -> <. <. V , W >. , K >. e. U_ x e. _V ( { x } X. ( 1st ` x ) ) )
7		fveq2 6191	. . . . 5 |- ( x = <. V , W >. -> ( 1st ` x ) = ( 1st ` <. V , W >. ) )
8	7	opeliunxp2 5260	. . . 4 |- ( <. <. V , W >. , K >. e. U_ x e. _V ( { x } X. ( 1st ` x ) ) <-> ( <. V , W >. e. _V /\ K e. ( 1st ` <. V , W >. ) ) )
9	8	simprbi 480	. . 3 |- ( <. <. V , W >. , K >. e. U_ x e. _V ( { x } X. ( 1st ` x ) ) -> K e. ( 1st ` <. V , W >. ) )
10	6, 9	syl 17	. 2 |- ( N e. ( <. V , W >. F K ) -> K e. ( 1st ` <. V , W >. ) )
11		op1stg 7180	. . 3 |- ( ( V e. X /\ W e. Y ) -> ( 1st ` <. V , W >. ) = V )
12	11	eleq2d 2687	. 2 |- ( ( V e. X /\ W e. Y ) -> ( K e. ( 1st ` <. V , W >. ) <-> K e. V ) )
13	10, 12	syl5ib 234	1 |- ( ( V e. X /\ W e. Y ) -> ( N e. ( <. V , W >. F K ) -> K e. V ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   <.cop 4183   U_ciun 4520   X. cxp 5112   dom cdm 5114   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169

This theorem is referenced by:  mpt2xopynvov0g  7340  mpt2xopovel  7346