Theorem fmpt2x 5846

Description: Functionality, domain and codomain of a class given by the "maps to" notation, where B ( x ) is not constant but depends on x. (Contributed by NM, 29-Dec-2014.)

Hypothesis

Ref	Expression
fmpt2x.1	|- F = ( x e. A , y e. B |-> C )

Assertion

Ref	Expression
fmpt2x	|- ( A. x e. A A. y e. B C e. D <-> F : U_ x e. A ( { x } X. B ) --> D )

Distinct variable groups:   x, y, A   y, B   x, D, y

Allowed substitution hints:   B( x)   C( x, y)   F( x, y)

Proof of Theorem fmpt2x

Dummy variables v w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2604	. . . . . . . 8 |- z e. _V
2		vex 2604	. . . . . . . 8 |- w e. _V
3	1, 2	op1std 5795	. . . . . . 7 |- ( v = <. z , w >. -> ( 1st ` v ) = z )
4	3	csbeq1d 2914	. . . . . 6 |- ( v = <. z , w >. -> [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C = [_ z / x ]_ [_ ( 2nd ` v ) / y ]_ C )
5	1, 2	op2ndd 5796	. . . . . . . 8 |- ( v = <. z , w >. -> ( 2nd ` v ) = w )
6	5	csbeq1d 2914	. . . . . . 7 |- ( v = <. z , w >. -> [_ ( 2nd ` v ) / y ]_ C = [_ w / y ]_ C )
7	6	csbeq2dv 2931	. . . . . 6 |- ( v = <. z , w >. -> [_ z / x ]_ [_ ( 2nd ` v ) / y ]_ C = [_ z / x ]_ [_ w / y ]_ C )
8	4, 7	eqtrd 2113	. . . . 5 |- ( v = <. z , w >. -> [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C = [_ z / x ]_ [_ w / y ]_ C )
9	8	eleq1d 2147	. . . 4 |- ( v = <. z , w >. -> ( [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C e. D <-> [_ z / x ]_ [_ w / y ]_ C e. D ) )
10	9	raliunxp 4495	. . 3 |- ( A. v e. U_ z e. A ( { z } X. [_ z / x ]_ B ) [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C e. D <-> A. z e. A A. w e. [_ z / x ]_ B [_ z / x ]_ [_ w / y ]_ C e. D )
11		nfv 1461	. . . . . . 7 |- F/ z ( ( x e. A /\ y e. B ) /\ v = C )
12		nfv 1461	. . . . . . 7 |- F/ w ( ( x e. A /\ y e. B ) /\ v = C )
13		nfv 1461	. . . . . . . . 9 |- F/ x z e. A
14		nfcsb1v 2938	. . . . . . . . . 10 |- F/_ x [_ z / x ]_ B
15	14	nfcri 2213	. . . . . . . . 9 |- F/ x w e. [_ z / x ]_ B
16	13, 15	nfan 1497	. . . . . . . 8 |- F/ x ( z e. A /\ w e. [_ z / x ]_ B )
17		nfcsb1v 2938	. . . . . . . . 9 |- F/_ x [_ z / x ]_ [_ w / y ]_ C
18	17	nfeq2 2230	. . . . . . . 8 |- F/ x v = [_ z / x ]_ [_ w / y ]_ C
19	16, 18	nfan 1497	. . . . . . 7 |- F/ x ( ( z e. A /\ w e. [_ z / x ]_ B ) /\ v = [_ z / x ]_ [_ w / y ]_ C )
20		nfv 1461	. . . . . . . 8 |- F/ y ( z e. A /\ w e. [_ z / x ]_ B )
21		nfcv 2219	. . . . . . . . . 10 |- F/_ y z
22		nfcsb1v 2938	. . . . . . . . . 10 |- F/_ y [_ w / y ]_ C
23	21, 22	nfcsb 2940	. . . . . . . . 9 |- F/_ y [_ z / x ]_ [_ w / y ]_ C
24	23	nfeq2 2230	. . . . . . . 8 |- F/ y v = [_ z / x ]_ [_ w / y ]_ C
25	20, 24	nfan 1497	. . . . . . 7 |- F/ y ( ( z e. A /\ w e. [_ z / x ]_ B ) /\ v = [_ z / x ]_ [_ w / y ]_ C )
26		eleq1 2141	. . . . . . . . . 10 |- ( x = z -> ( x e. A <-> z e. A ) )
27	26	adantr 270	. . . . . . . . 9 |- ( ( x = z /\ y = w ) -> ( x e. A <-> z e. A ) )
28		eleq1 2141	. . . . . . . . . 10 |- ( y = w -> ( y e. B <-> w e. B ) )
29		csbeq1a 2916	. . . . . . . . . . 11 |- ( x = z -> B = [_ z / x ]_ B )
30	29	eleq2d 2148	. . . . . . . . . 10 |- ( x = z -> ( w e. B <-> w e. [_ z / x ]_ B ) )
31	28, 30	sylan9bbr 450	. . . . . . . . 9 |- ( ( x = z /\ y = w ) -> ( y e. B <-> w e. [_ z / x ]_ B ) )
32	27, 31	anbi12d 456	. . . . . . . 8 |- ( ( x = z /\ y = w ) -> ( ( x e. A /\ y e. B ) <-> ( z e. A /\ w e. [_ z / x ]_ B ) ) )
33		csbeq1a 2916	. . . . . . . . . 10 |- ( y = w -> C = [_ w / y ]_ C )
34		csbeq1a 2916	. . . . . . . . . 10 |- ( x = z -> [_ w / y ]_ C = [_ z / x ]_ [_ w / y ]_ C )
35	33, 34	sylan9eqr 2135	. . . . . . . . 9 |- ( ( x = z /\ y = w ) -> C = [_ z / x ]_ [_ w / y ]_ C )
36	35	eqeq2d 2092	. . . . . . . 8 |- ( ( x = z /\ y = w ) -> ( v = C <-> v = [_ z / x ]_ [_ w / y ]_ C ) )
37	32, 36	anbi12d 456	. . . . . . 7 |- ( ( x = z /\ y = w ) -> ( ( ( x e. A /\ y e. B ) /\ v = C ) <-> ( ( z e. A /\ w e. [_ z / x ]_ B ) /\ v = [_ z / x ]_ [_ w / y ]_ C ) ) )
38	11, 12, 19, 25, 37	cbvoprab12 5598	. . . . . 6 |- { <. <. x , y >. , v >. | ( ( x e. A /\ y e. B ) /\ v = C ) } = { <. <. z , w >. , v >. | ( ( z e. A /\ w e. [_ z / x ]_ B ) /\ v = [_ z / x ]_ [_ w / y ]_ C ) }
39		df-mpt2 5537	. . . . . 6 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , v >. | ( ( x e. A /\ y e. B ) /\ v = C ) }
40		df-mpt2 5537	. . . . . 6 |- ( z e. A , w e. [_ z / x ]_ B |-> [_ z / x ]_ [_ w / y ]_ C ) = { <. <. z , w >. , v >. | ( ( z e. A /\ w e. [_ z / x ]_ B ) /\ v = [_ z / x ]_ [_ w / y ]_ C ) }
41	38, 39, 40	3eqtr4i 2111	. . . . 5 |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. [_ z / x ]_ B |-> [_ z / x ]_ [_ w / y ]_ C )
42		fmpt2x.1	. . . . 5 |- F = ( x e. A , y e. B |-> C )
43	8	mpt2mptx 5615	. . . . 5 |- ( v e. U_ z e. A ( { z } X. [_ z / x ]_ B ) |-> [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C ) = ( z e. A , w e. [_ z / x ]_ B |-> [_ z / x ]_ [_ w / y ]_ C )
44	41, 42, 43	3eqtr4i 2111	. . . 4 |- F = ( v e. U_ z e. A ( { z } X. [_ z / x ]_ B ) |-> [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C )
45	44	fmpt 5340	. . 3 |- ( A. v e. U_ z e. A ( { z } X. [_ z / x ]_ B ) [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C e. D <-> F : U_ z e. A ( { z } X. [_ z / x ]_ B ) --> D )
46	10, 45	bitr3i 184	. 2 |- ( A. z e. A A. w e. [_ z / x ]_ B [_ z / x ]_ [_ w / y ]_ C e. D <-> F : U_ z e. A ( { z } X. [_ z / x ]_ B ) --> D )
47		nfv 1461	. . 3 |- F/ z A. y e. B C e. D
48	17	nfel1 2229	. . . 4 |- F/ x [_ z / x ]_ [_ w / y ]_ C e. D
49	14, 48	nfralxy 2402	. . 3 |- F/ x A. w e. [_ z / x ]_ B [_ z / x ]_ [_ w / y ]_ C e. D
50		nfv 1461	. . . . 5 |- F/ w C e. D
51	22	nfel1 2229	. . . . 5 |- F/ y [_ w / y ]_ C e. D
52	33	eleq1d 2147	. . . . 5 |- ( y = w -> ( C e. D <-> [_ w / y ]_ C e. D ) )
53	50, 51, 52	cbvral 2573	. . . 4 |- ( A. y e. B C e. D <-> A. w e. B [_ w / y ]_ C e. D )
54	34	eleq1d 2147	. . . . 5 |- ( x = z -> ( [_ w / y ]_ C e. D <-> [_ z / x ]_ [_ w / y ]_ C e. D ) )
55	29, 54	raleqbidv 2561	. . . 4 |- ( x = z -> ( A. w e. B [_ w / y ]_ C e. D <-> A. w e. [_ z / x ]_ B [_ z / x ]_ [_ w / y ]_ C e. D ) )
56	53, 55	syl5bb 190	. . 3 |- ( x = z -> ( A. y e. B C e. D <-> A. w e. [_ z / x ]_ B [_ z / x ]_ [_ w / y ]_ C e. D ) )
57	47, 49, 56	cbvral 2573	. 2 |- ( A. x e. A A. y e. B C e. D <-> A. z e. A A. w e. [_ z / x ]_ B [_ z / x ]_ [_ w / y ]_ C e. D )
58		nfcv 2219	. . . 4 |- F/_ z ( { x } X. B )
59		nfcv 2219	. . . . 5 |- F/_ x { z }
60	59, 14	nfxp 4389	. . . 4 |- F/_ x ( { z } X. [_ z / x ]_ B )
61		sneq 3409	. . . . 5 |- ( x = z -> { x } = { z } )
62	61, 29	xpeq12d 4388	. . . 4 |- ( x = z -> ( { x } X. B ) = ( { z } X. [_ z / x ]_ B ) )
63	58, 60, 62	cbviun 3715	. . 3 |- U_ x e. A ( { x } X. B ) = U_ z e. A ( { z } X. [_ z / x ]_ B )
64	63	feq2i 5060	. 2 |- ( F : U_ x e. A ( { x } X. B ) --> D <-> F : U_ z e. A ( { z } X. [_ z / x ]_ B ) --> D )
65	46, 57, 64	3bitr4i 210	1 |- ( A. x e. A A. y e. B C e. D <-> F : U_ x e. A ( { x } X. B ) --> D )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348   [_csb 2908   {csn 3398   <.cop 3401   U_ciun 3678   |-> cmpt 3839   X. cxp 4361   -->wf 4918   ` cfv 4922   {coprab 5533   |-> cmpt2 5534   1stc1st 5785   2ndc2nd 5786

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-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

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  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-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  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-iun 3680  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-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788

This theorem is referenced by:  fmpt2  5847