Theorem fmpox 6436

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 2824	. . . . . . . 8 |- z e. _V
2		vex 2824	. . . . . . . 8 |- w e. _V
3	1, 2	op1std 6382	. . . . . . 7 |- ( v = <. z , w >. -> ( 1st ` v ) = z )
4	3	csbeq1d 3154	. . . . . 6 |- ( v = <. z , w >. -> [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C = [_ z / x ]_ [_ ( 2nd ` v ) / y ]_ C )
5	1, 2	op2ndd 6383	. . . . . . . 8 |- ( v = <. z , w >. -> ( 2nd ` v ) = w )
6	5	csbeq1d 3154	. . . . . . 7 |- ( v = <. z , w >. -> [_ ( 2nd ` v ) / y ]_ C = [_ w / y ]_ C )
7	6	csbeq2dv 3173	. . . . . 6 |- ( v = <. z , w >. -> [_ z / x ]_ [_ ( 2nd ` v ) / y ]_ C = [_ z / x ]_ [_ w / y ]_ C )
8	4, 7	eqtrd 2271	. . . . 5 |- ( v = <. z , w >. -> [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C = [_ z / x ]_ [_ w / y ]_ C )
9	8	eleq1d 2307	. . . 4 |- ( v = <. z , w >. -> ( [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C e. D <-> [_ z / x ]_ [_ w / y ]_ C e. D ) )
10	9	raliunxp 4921	. . 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 1581	. . . . . . 7 |- F/ z ( ( x e. A /\ y e. B ) /\ v = C )
12		nfv 1581	. . . . . . 7 |- F/ w ( ( x e. A /\ y e. B ) /\ v = C )
13		nfv 1581	. . . . . . . . 9 |- F/ x z e. A
14		nfcsb1v 3180	. . . . . . . . . 10 |- F/_ x [_ z / x ]_ B
15	14	nfcri 2386	. . . . . . . . 9 |- F/ x w e. [_ z / x ]_ B
16	13, 15	nfan 1618	. . . . . . . 8 |- F/ x ( z e. A /\ w e. [_ z / x ]_ B )
17		nfcsb1v 3180	. . . . . . . . 9 |- F/_ x [_ z / x ]_ [_ w / y ]_ C
18	17	nfeq2 2404	. . . . . . . 8 |- F/ x v = [_ z / x ]_ [_ w / y ]_ C
19	16, 18	nfan 1618	. . . . . . 7 |- F/ x ( ( z e. A /\ w e. [_ z / x ]_ B ) /\ v = [_ z / x ]_ [_ w / y ]_ C )
20		nfv 1581	. . . . . . . 8 |- F/ y ( z e. A /\ w e. [_ z / x ]_ B )
21		nfcv 2392	. . . . . . . . . 10 |- F/_ y z
22		nfcsb1v 3180	. . . . . . . . . 10 |- F/_ y [_ w / y ]_ C
23	21, 22	nfcsb 3185	. . . . . . . . 9 |- F/_ y [_ z / x ]_ [_ w / y ]_ C
24	23	nfeq2 2404	. . . . . . . 8 |- F/ y v = [_ z / x ]_ [_ w / y ]_ C
25	20, 24	nfan 1618	. . . . . . 7 |- F/ y ( ( z e. A /\ w e. [_ z / x ]_ B ) /\ v = [_ z / x ]_ [_ w / y ]_ C )
26		eleq1 2301	. . . . . . . . . 10 |- ( x = z -> ( x e. A <-> z e. A ) )
27	26	adantr 276	. . . . . . . . 9 |- ( ( x = z /\ y = w ) -> ( x e. A <-> z e. A ) )
28		eleq1 2301	. . . . . . . . . 10 |- ( y = w -> ( y e. B <-> w e. B ) )
29		csbeq1a 3156	. . . . . . . . . . 11 |- ( x = z -> B = [_ z / x ]_ B )
30	29	eleq2d 2308	. . . . . . . . . 10 |- ( x = z -> ( w e. B <-> w e. [_ z / x ]_ B ) )
31	28, 30	sylan9bbr 467	. . . . . . . . 9 |- ( ( x = z /\ y = w ) -> ( y e. B <-> w e. [_ z / x ]_ B ) )
32	27, 31	anbi12d 477	. . . . . . . 8 |- ( ( x = z /\ y = w ) -> ( ( x e. A /\ y e. B ) <-> ( z e. A /\ w e. [_ z / x ]_ B ) ) )
33		csbeq1a 3156	. . . . . . . . . 10 |- ( y = w -> C = [_ w / y ]_ C )
34		csbeq1a 3156	. . . . . . . . . 10 |- ( x = z -> [_ w / y ]_ C = [_ z / x ]_ [_ w / y ]_ C )
35	33, 34	sylan9eqr 2293	. . . . . . . . 9 |- ( ( x = z /\ y = w ) -> C = [_ z / x ]_ [_ w / y ]_ C )
36	35	eqeq2d 2250	. . . . . . . 8 |- ( ( x = z /\ y = w ) -> ( v = C <-> v = [_ z / x ]_ [_ w / y ]_ C ) )
37	32, 36	anbi12d 477	. . . . . . 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 6162	. . . . . 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 6090	. . . . . 6 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , v >. | ( ( x e. A /\ y e. B ) /\ v = C ) }
40		df-mpo 6090	. . . . . 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 2269	. . . . 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 6179	. . . . 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 2269	. . . 4 |- F = ( v e. U_ z e. A ( { z } X. [_ z / x ]_ B ) |-> [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C )
45	44	fmpt 5858	. . 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 186	. 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 1581	. . 3 |- F/ z A. y e. B C e. D
48	17	nfel1 2403	. . . 4 |- F/ x [_ z / x ]_ [_ w / y ]_ C e. D
49	14, 48	nfralxy 2588	. . 3 |- F/ x A. w e. [_ z / x ]_ B [_ z / x ]_ [_ w / y ]_ C e. D
50		nfv 1581	. . . . 5 |- F/ w C e. D
51	22	nfel1 2403	. . . . 5 |- F/ y [_ w / y ]_ C e. D
52	33	eleq1d 2307	. . . . 5 |- ( y = w -> ( C e. D <-> [_ w / y ]_ C e. D ) )
53	50, 51, 52	cbvral 2782	. . . 4 |- ( A. y e. B C e. D <-> A. w e. B [_ w / y ]_ C e. D )
54	34	eleq1d 2307	. . . . 5 |- ( x = z -> ( [_ w / y ]_ C e. D <-> [_ z / x ]_ [_ w / y ]_ C e. D ) )
55	29, 54	raleqbidv 2765	. . . 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	bitrid 192	. . 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 2782	. 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 2392	. . . 4 |- F/_ z ( { x } X. B )
59		nfcv 2392	. . . . 5 |- F/_ x { z }
60	59, 14	nfxp 4801	. . . 4 |- F/_ x ( { z } X. [_ z / x ]_ B )
61		sneq 3720	. . . . 5 |- ( x = z -> { x } = { z } )
62	61, 29	xpeq12d 4799	. . . 4 |- ( x = z -> ( { x } X. B ) = ( { z } X. [_ z / x ]_ B ) )
63	58, 60, 62	cbviun 4049	. . 3 |- U_ x e. A ( { x } X. B ) = U_ z e. A ( { z } X. [_ z / x ]_ B )
64	63	feq2i 5527	. 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 212	1 |- ( A. x e. A A. y e. B C e. D <-> F : U_ x e. A ( { x } X. B ) --> D )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   [_csb 3147   {csn 3709   <.cop 3712   U_ciun 4012   |-> cmpt 4192   X. cxp 4772   -->wf 5373   ` cfv 5377   {coprab 6086   e. cmpo 6087   1stc1st 6372   2ndc2nd 6373

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375

This theorem is used by:  fmpo  6437