Theorem fmpox 6179

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

Assertion

Ref	Expression
fmpox	|- ( 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 fmpox

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

Step	Hyp	Ref	Expression
1		vex 2733	. . . . . . . 8 |- z e. _V
2		vex 2733	. . . . . . . 8 |- w e. _V
3	1, 2	op1std 6127	. . . . . . 7 |- ( v = <. z , w >. -> ( 1st ` v ) = z )
4	3	csbeq1d 3056	. . . . . 6 |- ( v = <. z , w >. -> [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C = [_ z / x ]_ [_ ( 2nd ` v ) / y ]_ C )
5	1, 2	op2ndd 6128	. . . . . . . 8 |- ( v = <. z , w >. -> ( 2nd ` v ) = w )
6	5	csbeq1d 3056	. . . . . . 7 |- ( v = <. z , w >. -> [_ ( 2nd ` v ) / y ]_ C = [_ w / y ]_ C )
7	6	csbeq2dv 3075	. . . . . 6 |- ( v = <. z , w >. -> [_ z / x ]_ [_ ( 2nd ` v ) / y ]_ C = [_ z / x ]_ [_ w / y ]_ C )
8	4, 7	eqtrd 2203	. . . . 5 |- ( v = <. z , w >. -> [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C = [_ z / x ]_ [_ w / y ]_ C )
9	8	eleq1d 2239	. . . 4 |- ( v = <. z , w >. -> ( [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C e. D <-> [_ z / x ]_ [_ w / y ]_ C e. D ) )
10	9	raliunxp 4752	. . 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 1521	. . . . . . 7 |- F/ z ( ( x e. A /\ y e. B ) /\ v = C )
12		nfv 1521	. . . . . . 7 |- F/ w ( ( x e. A /\ y e. B ) /\ v = C )
13		nfv 1521	. . . . . . . . 9 |- F/ x z e. A
14		nfcsb1v 3082	. . . . . . . . . 10 |- F/_ x [_ z / x ]_ B
15	14	nfcri 2306	. . . . . . . . 9 |- F/ x w e. [_ z / x ]_ B
16	13, 15	nfan 1558	. . . . . . . 8 |- F/ x ( z e. A /\ w e. [_ z / x ]_ B )
17		nfcsb1v 3082	. . . . . . . . 9 |- F/_ x [_ z / x ]_ [_ w / y ]_ C
18	17	nfeq2 2324	. . . . . . . 8 |- F/ x v = [_ z / x ]_ [_ w / y ]_ C
19	16, 18	nfan 1558	. . . . . . 7 |- F/ x ( ( z e. A /\ w e. [_ z / x ]_ B ) /\ v = [_ z / x ]_ [_ w / y ]_ C )
20		nfv 1521	. . . . . . . 8 |- F/ y ( z e. A /\ w e. [_ z / x ]_ B )
21		nfcv 2312	. . . . . . . . . 10 |- F/_ y z
22		nfcsb1v 3082	. . . . . . . . . 10 |- F/_ y [_ w / y ]_ C
23	21, 22	nfcsb 3086	. . . . . . . . 9 |- F/_ y [_ z / x ]_ [_ w / y ]_ C
24	23	nfeq2 2324	. . . . . . . 8 |- F/ y v = [_ z / x ]_ [_ w / y ]_ C
25	20, 24	nfan 1558	. . . . . . 7 |- F/ y ( ( z e. A /\ w e. [_ z / x ]_ B ) /\ v = [_ z / x ]_ [_ w / y ]_ C )
26		eleq1 2233	. . . . . . . . . 10 |- ( x = z -> ( x e. A <-> z e. A ) )
27	26	adantr 274	. . . . . . . . 9 |- ( ( x = z /\ y = w ) -> ( x e. A <-> z e. A ) )
28		eleq1 2233	. . . . . . . . . 10 |- ( y = w -> ( y e. B <-> w e. B ) )
29		csbeq1a 3058	. . . . . . . . . . 11 |- ( x = z -> B = [_ z / x ]_ B )
30	29	eleq2d 2240	. . . . . . . . . 10 |- ( x = z -> ( w e. B <-> w e. [_ z / x ]_ B ) )
31	28, 30	sylan9bbr 460	. . . . . . . . 9 |- ( ( x = z /\ y = w ) -> ( y e. B <-> w e. [_ z / x ]_ B ) )
32	27, 31	anbi12d 470	. . . . . . . 8 |- ( ( x = z /\ y = w ) -> ( ( x e. A /\ y e. B ) <-> ( z e. A /\ w e. [_ z / x ]_ B ) ) )
33		csbeq1a 3058	. . . . . . . . . 10 |- ( y = w -> C = [_ w / y ]_ C )
34		csbeq1a 3058	. . . . . . . . . 10 |- ( x = z -> [_ w / y ]_ C = [_ z / x ]_ [_ w / y ]_ C )
35	33, 34	sylan9eqr 2225	. . . . . . . . 9 |- ( ( x = z /\ y = w ) -> C = [_ z / x ]_ [_ w / y ]_ C )
36	35	eqeq2d 2182	. . . . . . . 8 |- ( ( x = z /\ y = w ) -> ( v = C <-> v = [_ z / x ]_ [_ w / y ]_ C ) )
37	32, 36	anbi12d 470	. . . . . . 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 5927	. . . . . 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-mpo 5858	. . . . . 6 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , v >. | ( ( x e. A /\ y e. B ) /\ v = C ) }
40		df-mpo 5858	. . . . . 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 2201	. . . . 5 |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. [_ z / x ]_ B |-> [_ z / x ]_ [_ w / y ]_ C )
42		fmpox.1	. . . . 5 |- F = ( x e. A , y e. B |-> C )
43	8	mpomptx 5944	. . . . 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 2201	. . . 4 |- F = ( v e. U_ z e. A ( { z } X. [_ z / x ]_ B ) |-> [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C )
45	44	fmpt 5646	. . 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 185	. 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 1521	. . 3 |- F/ z A. y e. B C e. D
48	17	nfel1 2323	. . . 4 |- F/ x [_ z / x ]_ [_ w / y ]_ C e. D
49	14, 48	nfralxy 2508	. . 3 |- F/ x A. w e. [_ z / x ]_ B [_ z / x ]_ [_ w / y ]_ C e. D
50		nfv 1521	. . . . 5 |- F/ w C e. D
51	22	nfel1 2323	. . . . 5 |- F/ y [_ w / y ]_ C e. D
52	33	eleq1d 2239	. . . . 5 |- ( y = w -> ( C e. D <-> [_ w / y ]_ C e. D ) )
53	50, 51, 52	cbvral 2692	. . . 4 |- ( A. y e. B C e. D <-> A. w e. B [_ w / y ]_ C e. D )
54	34	eleq1d 2239	. . . . 5 |- ( x = z -> ( [_ w / y ]_ C e. D <-> [_ z / x ]_ [_ w / y ]_ C e. D ) )
55	29, 54	raleqbidv 2677	. . . 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 191	. . 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 2692	. 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 2312	. . . 4 |- F/_ z ( { x } X. B )
59		nfcv 2312	. . . . 5 |- F/_ x { z }
60	59, 14	nfxp 4638	. . . 4 |- F/_ x ( { z } X. [_ z / x ]_ B )
61		sneq 3594	. . . . 5 |- ( x = z -> { x } = { z } )
62	61, 29	xpeq12d 4636	. . . 4 |- ( x = z -> ( { x } X. B ) = ( { z } X. [_ z / x ]_ B ) )
63	58, 60, 62	cbviun 3910	. . 3 |- U_ x e. A ( { x } X. B ) = U_ z e. A ( { z } X. [_ z / x ]_ B )
64	63	feq2i 5341	. 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 211	1 |- ( A. x e. A A. y e. B C e. D <-> F : U_ x e. A ( { x } X. B ) --> D )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   A.wral 2448   [_csb 3049   {csn 3583   <.cop 3586   U_ciun 3873   |-> cmpt 4050   X. cxp 4609   -->wf 5194   ` cfv 5198   {coprab 5854   e. cmpo 5855   1stc1st 6117   2ndc2nd 6118

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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120

This theorem is referenced by:  fmpo  6180