Theorem mpt2xopoveq 6005

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 5305	. . . . 5 |- ( x = <. V , W >. -> ( 1st ` x ) = ( 1st ` <. V , W >. ) )
4		op1stg 5921	. . . . . 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 2142	. . . 4 |- ( ( ( ( V e. X /\ W e. Y ) /\ K e. V ) /\ x = <. V , W >. ) -> ( 1st ` x ) = V )
7	6	adantrr 463	. . 3 |- ( ( ( ( V e. X /\ W e. Y ) /\ K e. V ) /\ ( x = <. V , W >. /\ y = K ) ) -> ( 1st ` x ) = V )
8		sbceq1a 2849	. . . . . 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 2849	. . . . . 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 2614	. 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 4055	. . 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 3982	. . 3 |- ( V e. X -> { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } e. _V )
20	19	ad2antrr 472	. 2 |- ( ( ( V e. X /\ W e. Y ) /\ K e. V ) -> { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } e. _V )
21		equid 1634	. . 3 |- z = z
22		nfvd 1467	. . 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 1467	. . 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 2228	. 2 |- F/_ y <. V , W >.
27		nfcv 2228	. 2 |- F/_ x K
28		nfsbc1v 2858	. . 3 |- F/ x [. <. V , W >. / x ]. [. K / y ]. ph
29		nfcv 2228	. . 3 |- F/_ x V
30	28, 29	nfrabxy 2547	. 2 |- F/_ x { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph }
31		nfsbc1v 2858	. . . 4 |- F/ y [. K / y ]. ph
32	26, 31	nfsbc 2860	. . 3 |- F/ y [. <. V , W >. / x ]. [. K / y ]. ph
33		nfcv 2228	. . 3 |- F/_ y V
34	32, 33	nfrabxy 2547	. 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 5770	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 1289   F/wnf 1394   e. wcel 1438   {crab 2363   _Vcvv 2619   [.wsbc 2840   <.cop 3449   ` cfv 5015  (class class class)co 5652   |-> cmpt2 5654   1stc1st 5909

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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911

This theorem is referenced by:  mpt2xopovel  6006