Theorem mpoxopoveq 6307

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 5561	. . . . 5 |- ( x = <. V , W >. -> ( 1st ` x ) = ( 1st ` <. V , W >. ) )
4		op1stg 6217	. . . . . 6 |- ( ( V e. X /\ W e. Y ) -> ( 1st ` <. V , W >. ) = V )
5	4	adantr 276	. . . . 5 |- ( ( ( V e. X /\ W e. Y ) /\ K e. V ) -> ( 1st ` <. V , W >. ) = V )
6	3, 5	sylan9eqr 2251	. . . 4 |- ( ( ( ( V e. X /\ W e. Y ) /\ K e. V ) /\ x = <. V , W >. ) -> ( 1st ` x ) = V )
7	6	adantrr 479	. . 3 |- ( ( ( ( V e. X /\ W e. Y ) /\ K e. V ) /\ ( x = <. V , W >. /\ y = K ) ) -> ( 1st ` x ) = V )
8		sbceq1a 2999	. . . . . 6 |- ( y = K -> ( ph <-> [. K / y ]. ph ) )
9	8	adantl 277	. . . . 5 |- ( ( x = <. V , W >. /\ y = K ) -> ( ph <-> [. K / y ]. ph ) )
10	9	adantl 277	. . . 4 |- ( ( ( ( V e. X /\ W e. Y ) /\ K e. V ) /\ ( x = <. V , W >. /\ y = K ) ) -> ( ph <-> [. K / y ]. ph ) )
11		sbceq1a 2999	. . . . . 6 |- ( x = <. V , W >. -> ( [. K / y ]. ph <-> [. <. V , W >. / x ]. [. K / y ]. ph ) )
12	11	adantr 276	. . . . 5 |- ( ( x = <. V , W >. /\ y = K ) -> ( [. K / y ]. ph <-> [. <. V , W >. / x ]. [. K / y ]. ph ) )
13	12	adantl 277	. . . 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 188	. . 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 2758	. 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 4262	. . 3 |- ( ( V e. X /\ W e. Y ) -> <. V , W >. e. _V )
17	16	adantr 276	. 2 |- ( ( ( V e. X /\ W e. Y ) /\ K e. V ) -> <. V , W >. e. _V )
18		simpr 110	. 2 |- ( ( ( V e. X /\ W e. Y ) /\ K e. V ) -> K e. V )
19		rabexg 4177	. . 3 |- ( V e. X -> { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } e. _V )
20	19	ad2antrr 488	. 2 |- ( ( ( V e. X /\ W e. Y ) /\ K e. V ) -> { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } e. _V )
21		equid 1715	. . 3 |- z = z
22		nfvd 1543	. . 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 1543	. . 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 2339	. 2 |- F/_ y <. V , W >.
27		nfcv 2339	. 2 |- F/_ x K
28		nfsbc1v 3008	. . 3 |- F/ x [. <. V , W >. / x ]. [. K / y ]. ph
29		nfcv 2339	. . 3 |- F/_ x V
30	28, 29	nfrabw 2678	. 2 |- F/_ x { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph }
31		nfsbc1v 3008	. . . 4 |- F/ y [. K / y ]. ph
32	26, 31	nfsbc 3010	. . 3 |- F/ y [. <. V , W >. / x ]. [. K / y ]. ph
33		nfcv 2339	. . 3 |- F/_ y V
34	32, 33	nfrabw 2678	. 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 6052	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 104   <-> wb 105   = wceq 1364   F/wnf 1474   e. wcel 2167   {crab 2479   _Vcvv 2763   [.wsbc 2989   <.cop 3626   ` cfv 5259  (class class class)co 5925   e. cmpo 5927   1stc1st 6205

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-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-iota 5220  df-fun 5261  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207

This theorem is referenced by:  mpoxopovel  6308