Theorem mpoxopoveq 6177

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

Assertion

Ref	Expression
mpoxopoveq	|- ( ( ( 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 mpoxopoveq

Step	Hyp	Ref	Expression
1		mpoxopoveq.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 5461	. . . . 5 |- ( x = <. V , W >. -> ( 1st ` x ) = ( 1st ` <. V , W >. ) )
4		op1stg 6088	. . . . . 6 |- ( ( V e. X /\ W e. Y ) -> ( 1st ` <. V , W >. ) = V )
5	4	adantr 274	. . . . 5 |- ( ( ( V e. X /\ W e. Y ) /\ K e. V ) -> ( 1st ` <. V , W >. ) = V )
6	3, 5	sylan9eqr 2209	. . . 4 |- ( ( ( ( V e. X /\ W e. Y ) /\ K e. V ) /\ x = <. V , W >. ) -> ( 1st ` x ) = V )
7	6	adantrr 471	. . 3 |- ( ( ( ( V e. X /\ W e. Y ) /\ K e. V ) /\ ( x = <. V , W >. /\ y = K ) ) -> ( 1st ` x ) = V )
8		sbceq1a 2942	. . . . . 6 |- ( y = K -> ( ph <-> [. K / y ]. ph ) )
9	8	adantl 275	. . . . 5 |- ( ( x = <. V , W >. /\ y = K ) -> ( ph <-> [. K / y ]. ph ) )
10	9	adantl 275	. . . 4 |- ( ( ( ( V e. X /\ W e. Y ) /\ K e. V ) /\ ( x = <. V , W >. /\ y = K ) ) -> ( ph <-> [. K / y ]. ph ) )
11		sbceq1a 2942	. . . . . 6 |- ( x = <. V , W >. -> ( [. K / y ]. ph <-> [. <. V , W >. / x ]. [. K / y ]. ph ) )
12	11	adantr 274	. . . . 5 |- ( ( x = <. V , W >. /\ y = K ) -> ( [. K / y ]. ph <-> [. <. V , W >. / x ]. [. K / y ]. ph ) )
13	12	adantl 275	. . . 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 187	. . 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 2704	. 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 4183	. . 3 |- ( ( V e. X /\ W e. Y ) -> <. V , W >. e. _V )
17	16	adantr 274	. 2 |- ( ( ( V e. X /\ W e. Y ) /\ K e. V ) -> <. V , W >. e. _V )
18		simpr 109	. 2 |- ( ( ( V e. X /\ W e. Y ) /\ K e. V ) -> K e. V )
19		rabexg 4103	. . 3 |- ( V e. X -> { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } e. _V )
20	19	ad2antrr 480	. 2 |- ( ( ( V e. X /\ W e. Y ) /\ K e. V ) -> { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } e. _V )
21		equid 1678	. . 3 |- z = z
22		nfvd 1506	. . 3 |- ( z = z -> F/ x ( ( V e. X /\ W e. Y ) /\ K e. V ) )
23	21, 22	ax-mp 5	. 2 |- F/ x ( ( V e. X /\ W e. Y ) /\ K e. V )
24		nfvd 1506	. . 3 |- ( z = z -> F/ y ( ( V e. X /\ W e. Y ) /\ K e. V ) )
25	21, 24	ax-mp 5	. 2 |- F/ y ( ( V e. X /\ W e. Y ) /\ K e. V )
26		nfcv 2296	. 2 |- F/_ y <. V , W >.
27		nfcv 2296	. 2 |- F/_ x K
28		nfsbc1v 2951	. . 3 |- F/ x [. <. V , W >. / x ]. [. K / y ]. ph
29		nfcv 2296	. . 3 |- F/_ x V
30	28, 29	nfrabxy 2634	. 2 |- F/_ x { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph }
31		nfsbc1v 2951	. . . 4 |- F/ y [. K / y ]. ph
32	26, 31	nfsbc 2953	. . 3 |- F/ y [. <. V , W >. / x ]. [. K / y ]. ph
33		nfcv 2296	. . 3 |- F/_ y V
34	32, 33	nfrabxy 2634	. 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	ovmpodxf 5936	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 103   <-> wb 104   = wceq 1332   F/wnf 1437   e. wcel 2125   {crab 2436   _Vcvv 2709   [.wsbc 2933   <.cop 3559   ` cfv 5163  (class class class)co 5814   e. cmpo 5816   1stc1st 6076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-ral 2437  df-rex 2438  df-rab 2441  df-v 2711  df-sbc 2934  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-iota 5128  df-fun 5165  df-fv 5171  df-ov 5817  df-oprab 5818  df-mpo 5819  df-1st 6078

This theorem is referenced by:  mpoxopovel  6178