Theorem mpt2xopoveq 5878

Description: 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. (Contributed by Alexander van der Vekens, 11-Oct-2017.)

Hypothesis

Ref	Expression
mpt2xopoveq.f	|- F = ( x e. _V , y e. ( 1st ` x ) |-> { n e. ( 1st ` x ) | ph } )

Assertion

Ref	Expression
mpt2xopoveq	|- ( ( ( V e. X /\ W e. Y ) /\ K e. V ) -> ( <. V , W >. F K ) = { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } )

Distinct variable groups:   n, K, x, y   n, V, x, y   n, W, x, y   n, X, x, y   n, Y, x, y

Allowed substitution hints:   ph( x, y, n)   F( x, y, n)

Proof of Theorem mpt2xopoveq

Step	Hyp	Ref	Expression
1		mpt2xopoveq.f	. . 3 |- F = ( x e. _V , y e. ( 1st ` x ) |-> { n e. ( 1st ` x ) | ph } )
2	1	a1i 9	. 2 |- ( ( ( V e. X /\ W e. Y ) /\ K e. V ) -> F = ( x e. _V , y e. ( 1st ` x ) |-> { n e. ( 1st ` x ) | ph } ) )
3		fveq2 5198	. . . . 5 |- ( x = <. V , W >. -> ( 1st ` x ) = ( 1st ` <. V , W >. ) )
4		op1stg 5797	. . . . . 6 |- ( ( V e. X /\ W e. Y ) -> ( 1st ` <. V , W >. ) = V )
5	4	adantr 270	. . . . 5 |- ( ( ( V e. X /\ W e. Y ) /\ K e. V ) -> ( 1st ` <. V , W >. ) = V )
6	3, 5	sylan9eqr 2135	. . . 4 |- ( ( ( ( V e. X /\ W e. Y ) /\ K e. V ) /\ x = <. V , W >. ) -> ( 1st ` x ) = V )
7	6	adantrr 462	. . 3 |- ( ( ( ( V e. X /\ W e. Y ) /\ K e. V ) /\ ( x = <. V , W >. /\ y = K ) ) -> ( 1st ` x ) = V )
8		sbceq1a 2824	. . . . . 6 |- ( y = K -> ( ph <-> [. K / y ]. ph ) )
9	8	adantl 271	. . . . 5 |- ( ( x = <. V , W >. /\ y = K ) -> ( ph <-> [. K / y ]. ph ) )
10	9	adantl 271	. . . 4 |- ( ( ( ( V e. X /\ W e. Y ) /\ K e. V ) /\ ( x = <. V , W >. /\ y = K ) ) -> ( ph <-> [. K / y ]. ph ) )
11		sbceq1a 2824	. . . . . 6 |- ( x = <. V , W >. -> ( [. K / y ]. ph <-> [. <. V , W >. / x ]. [. K / y ]. ph ) )
12	11	adantr 270	. . . . 5 |- ( ( x = <. V , W >. /\ y = K ) -> ( [. K / y ]. ph <-> [. <. V , W >. / x ]. [. K / y ]. ph ) )
13	12	adantl 271	. . . 4 |- ( ( ( ( V e. X /\ W e. Y ) /\ K e. V ) /\ ( x = <. V , W >. /\ y = K ) ) -> ( [. K / y ]. ph <-> [. <. V , W >. / x ]. [. K / y ]. ph ) )
14	10, 13	bitrd 186	. . 3 |- ( ( ( ( V e. X /\ W e. Y ) /\ K e. V ) /\ ( x = <. V , W >. /\ y = K ) ) -> ( ph <-> [. <. V , W >. / x ]. [. K / y ]. ph ) )
15	7, 14	rabeqbidv 2596	. 2 |- ( ( ( ( V e. X /\ W e. Y ) /\ K e. V ) /\ ( x = <. V , W >. /\ y = K ) ) -> { n e. ( 1st ` x ) | ph } = { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } )
16		opexg 3983	. . 3 |- ( ( V e. X /\ W e. Y ) -> <. V , W >. e. _V )
17	16	adantr 270	. 2 |- ( ( ( V e. X /\ W e. Y ) /\ K e. V ) -> <. V , W >. e. _V )
18		simpr 108	. 2 |- ( ( ( V e. X /\ W e. Y ) /\ K e. V ) -> K e. V )
19		rabexg 3921	. . 3 |- ( V e. X -> { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } e. _V )
20	19	ad2antrr 471	. 2 |- ( ( ( V e. X /\ W e. Y ) /\ K e. V ) -> { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } e. _V )
21		equid 1629	. . 3 |- z = z
22		nfvd 1462	. . 3 |- ( z = z -> F/ x ( ( V e. X /\ W e. Y ) /\ K e. V ) )
23	21, 22	ax-mp 7	. 2 |- F/ x ( ( V e. X /\ W e. Y ) /\ K e. V )
24		nfvd 1462	. . 3 |- ( z = z -> F/ y ( ( V e. X /\ W e. Y ) /\ K e. V ) )
25	21, 24	ax-mp 7	. 2 |- F/ y ( ( V e. X /\ W e. Y ) /\ K e. V )
26		nfcv 2219	. 2 |- F/_ y <. V , W >.
27		nfcv 2219	. 2 |- F/_ x K
28		nfsbc1v 2833	. . 3 |- F/ x [. <. V , W >. / x ]. [. K / y ]. ph
29		nfcv 2219	. . 3 |- F/_ x V
30	28, 29	nfrabxy 2534	. 2 |- F/_ x { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph }
31		nfsbc1v 2833	. . . 4 |- F/ y [. K / y ]. ph
32	26, 31	nfsbc 2835	. . 3 |- F/ y [. <. V , W >. / x ]. [. K / y ]. ph
33		nfcv 2219	. . 3 |- F/_ y V
34	32, 33	nfrabxy 2534	. 2 |- F/_ y { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph }
35	2, 15, 6, 17, 18, 20, 23, 25, 26, 27, 30, 34	ovmpt2dxf 5646	1 |- ( ( ( V e. X /\ W e. Y ) /\ K e. V ) -> ( <. V , W >. F K ) = { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   F/wnf 1389   e. wcel 1433   {crab 2352   _Vcvv 2601   [.wsbc 2815   <.cop 3401   ` 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-in1 576  ax-in2 577  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  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  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-ne 2246  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  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-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-iota 4887  df-fun 4924  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787

This theorem is referenced by:  mpt2xopovel  5879