Theorem fmpt2x 5826

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 2560	. . . . . . . 8 |- z e. _V
2		vex 2560	. . . . . . . 8 |- w e. _V
3	1, 2	op1std 5775	. . . . . . 7 |- ( v = <. z , w >. -> ( 1st ` v ) = z )
4	3	csbeq1d 2858	. . . . . 6 |- ( v = <. z , w >. -> [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C = [_ z / x ]_ [_ ( 2nd ` v ) / y ]_ C )
5	1, 2	op2ndd 5776	. . . . . . . 8 |- ( v = <. z , w >. -> ( 2nd ` v ) = w )
6	5	csbeq1d 2858	. . . . . . 7 |- ( v = <. z , w >. -> [_ ( 2nd ` v ) / y ]_ C = [_ w / y ]_ C )
7	6	csbeq2dv 2875	. . . . . 6 |- ( v = <. z , w >. -> [_ z / x ]_ [_ ( 2nd ` v ) / y ]_ C = [_ z / x ]_ [_ w / y ]_ C )
8	4, 7	eqtrd 2072	. . . . 5 |- ( v = <. z , w >. -> [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C = [_ z / x ]_ [_ w / y ]_ C )
9	8	eleq1d 2106	. . . 4 |- ( v = <. z , w >. -> ( [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C e. D <-> [_ z / x ]_ [_ w / y ]_ C e. D ) )
10	9	raliunxp 4477	. . 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 1421	. . . . . . 7 |- F/ z ( ( x e. A /\ y e. B ) /\ v = C )
12		nfv 1421	. . . . . . 7 |- F/ w ( ( x e. A /\ y e. B ) /\ v = C )
13		nfv 1421	. . . . . . . . 9 |- F/ x z e. A
14		nfcsb1v 2882	. . . . . . . . . 10 |- F/_ x [_ z / x ]_ B
15	14	nfcri 2172	. . . . . . . . 9 |- F/ x w e. [_ z / x ]_ B
16	13, 15	nfan 1457	. . . . . . . 8 |- F/ x ( z e. A /\ w e. [_ z / x ]_ B )
17		nfcsb1v 2882	. . . . . . . . 9 |- F/_ x [_ z / x ]_ [_ w / y ]_ C
18	17	nfeq2 2189	. . . . . . . 8 |- F/ x v = [_ z / x ]_ [_ w / y ]_ C
19	16, 18	nfan 1457	. . . . . . 7 |- F/ x ( ( z e. A /\ w e. [_ z / x ]_ B ) /\ v = [_ z / x ]_ [_ w / y ]_ C )
20		nfv 1421	. . . . . . . 8 |- F/ y ( z e. A /\ w e. [_ z / x ]_ B )
21		nfcv 2178	. . . . . . . . . 10 |- F/_ y z
22		nfcsb1v 2882	. . . . . . . . . 10 |- F/_ y [_ w / y ]_ C
23	21, 22	nfcsb 2884	. . . . . . . . 9 |- F/_ y [_ z / x ]_ [_ w / y ]_ C
24	23	nfeq2 2189	. . . . . . . 8 |- F/ y v = [_ z / x ]_ [_ w / y ]_ C
25	20, 24	nfan 1457	. . . . . . 7 |- F/ y ( ( z e. A /\ w e. [_ z / x ]_ B ) /\ v = [_ z / x ]_ [_ w / y ]_ C )
26		eleq1 2100	. . . . . . . . . 10 |- ( x = z -> ( x e. A <-> z e. A ) )
27	26	adantr 261	. . . . . . . . 9 |- ( ( x = z /\ y = w ) -> ( x e. A <-> z e. A ) )
28		eleq1 2100	. . . . . . . . . 10 |- ( y = w -> ( y e. B <-> w e. B ) )
29		csbeq1a 2860	. . . . . . . . . . 11 |- ( x = z -> B = [_ z / x ]_ B )
30	29	eleq2d 2107	. . . . . . . . . 10 |- ( x = z -> ( w e. B <-> w e. [_ z / x ]_ B ) )
31	28, 30	sylan9bbr 436	. . . . . . . . 9 |- ( ( x = z /\ y = w ) -> ( y e. B <-> w e. [_ z / x ]_ B ) )
32	27, 31	anbi12d 442	. . . . . . . 8 |- ( ( x = z /\ y = w ) -> ( ( x e. A /\ y e. B ) <-> ( z e. A /\ w e. [_ z / x ]_ B ) ) )
33		csbeq1a 2860	. . . . . . . . . 10 |- ( y = w -> C = [_ w / y ]_ C )
34		csbeq1a 2860	. . . . . . . . . 10 |- ( x = z -> [_ w / y ]_ C = [_ z / x ]_ [_ w / y ]_ C )
35	33, 34	sylan9eqr 2094	. . . . . . . . 9 |- ( ( x = z /\ y = w ) -> C = [_ z / x ]_ [_ w / y ]_ C )
36	35	eqeq2d 2051	. . . . . . . 8 |- ( ( x = z /\ y = w ) -> ( v = C <-> v = [_ z / x ]_ [_ w / y ]_ C ) )
37	32, 36	anbi12d 442	. . . . . . 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 5578	. . . . . 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 5517	. . . . . 6 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , v >. | ( ( x e. A /\ y e. B ) /\ v = C ) }
40		df-mpt2 5517	. . . . . 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 2070	. . . . 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 5595	. . . . 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 2070	. . . 4 |- F = ( v e. U_ z e. A ( { z } X. [_ z / x ]_ B ) |-> [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C )
45	44	fmpt 5319	. . 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 175	. 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 1421	. . 3 |- F/ z A. y e. B C e. D
48	17	nfel1 2188	. . . 4 |- F/ x [_ z / x ]_ [_ w / y ]_ C e. D
49	14, 48	nfralxy 2360	. . 3 |- F/ x A. w e. [_ z / x ]_ B [_ z / x ]_ [_ w / y ]_ C e. D
50		nfv 1421	. . . . 5 |- F/ w C e. D
51	22	nfel1 2188	. . . . 5 |- F/ y [_ w / y ]_ C e. D
52	33	eleq1d 2106	. . . . 5 |- ( y = w -> ( C e. D <-> [_ w / y ]_ C e. D ) )
53	50, 51, 52	cbvral 2529	. . . 4 |- ( A. y e. B C e. D <-> A. w e. B [_ w / y ]_ C e. D )
54	34	eleq1d 2106	. . . . 5 |- ( x = z -> ( [_ w / y ]_ C e. D <-> [_ z / x ]_ [_ w / y ]_ C e. D ) )
55	29, 54	raleqbidv 2517	. . . 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 181	. . 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 2529	. 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 2178	. . . 4 |- F/_ z ( { x } X. B )
59		nfcv 2178	. . . . 5 |- F/_ x { z }
60	59, 14	nfxp 4371	. . . 4 |- F/_ x ( { z } X. [_ z / x ]_ B )
61		sneq 3386	. . . . 5 |- ( x = z -> { x } = { z } )
62	61, 29	xpeq12d 4370	. . . 4 |- ( x = z -> ( { x } X. B ) = ( { z } X. [_ z / x ]_ B ) )
63	58, 60, 62	cbviun 3694	. . 3 |- U_ x e. A ( { x } X. B ) = U_ z e. A ( { z } X. [_ z / x ]_ B )
64	63	feq2i 5040	. 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 201	1 |- ( A. x e. A A. y e. B C e. D <-> F : U_ x e. A ( { x } X. B ) --> D )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   A.wral 2306   [_csb 2852   {csn 3375   <.cop 3378   U_ciun 3657   |-> cmpt 3818   X. cxp 4343   -->wf 4898   ` cfv 4902   {coprab 5513   |-> cmpt2 5514   1stc1st 5765   2ndc2nd 5766

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768

This theorem is referenced by:  fmpt2  5827