Theorem fmpox 6106

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 2692	. . . . . . . 8 |- z e. _V
2		vex 2692	. . . . . . . 8 |- w e. _V
3	1, 2	op1std 6054	. . . . . . 7 |- ( v = <. z , w >. -> ( 1st ` v ) = z )
4	3	csbeq1d 3014	. . . . . 6 |- ( v = <. z , w >. -> [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C = [_ z / x ]_ [_ ( 2nd ` v ) / y ]_ C )
5	1, 2	op2ndd 6055	. . . . . . . 8 |- ( v = <. z , w >. -> ( 2nd ` v ) = w )
6	5	csbeq1d 3014	. . . . . . 7 |- ( v = <. z , w >. -> [_ ( 2nd ` v ) / y ]_ C = [_ w / y ]_ C )
7	6	csbeq2dv 3033	. . . . . 6 |- ( v = <. z , w >. -> [_ z / x ]_ [_ ( 2nd ` v ) / y ]_ C = [_ z / x ]_ [_ w / y ]_ C )
8	4, 7	eqtrd 2173	. . . . 5 |- ( v = <. z , w >. -> [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C = [_ z / x ]_ [_ w / y ]_ C )
9	8	eleq1d 2209	. . . 4 |- ( v = <. z , w >. -> ( [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C e. D <-> [_ z / x ]_ [_ w / y ]_ C e. D ) )
10	9	raliunxp 4688	. . 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 1509	. . . . . . 7 |- F/ z ( ( x e. A /\ y e. B ) /\ v = C )
12		nfv 1509	. . . . . . 7 |- F/ w ( ( x e. A /\ y e. B ) /\ v = C )
13		nfv 1509	. . . . . . . . 9 |- F/ x z e. A
14		nfcsb1v 3040	. . . . . . . . . 10 |- F/_ x [_ z / x ]_ B
15	14	nfcri 2276	. . . . . . . . 9 |- F/ x w e. [_ z / x ]_ B
16	13, 15	nfan 1545	. . . . . . . 8 |- F/ x ( z e. A /\ w e. [_ z / x ]_ B )
17		nfcsb1v 3040	. . . . . . . . 9 |- F/_ x [_ z / x ]_ [_ w / y ]_ C
18	17	nfeq2 2294	. . . . . . . 8 |- F/ x v = [_ z / x ]_ [_ w / y ]_ C
19	16, 18	nfan 1545	. . . . . . 7 |- F/ x ( ( z e. A /\ w e. [_ z / x ]_ B ) /\ v = [_ z / x ]_ [_ w / y ]_ C )
20		nfv 1509	. . . . . . . 8 |- F/ y ( z e. A /\ w e. [_ z / x ]_ B )
21		nfcv 2282	. . . . . . . . . 10 |- F/_ y z
22		nfcsb1v 3040	. . . . . . . . . 10 |- F/_ y [_ w / y ]_ C
23	21, 22	nfcsb 3042	. . . . . . . . 9 |- F/_ y [_ z / x ]_ [_ w / y ]_ C
24	23	nfeq2 2294	. . . . . . . 8 |- F/ y v = [_ z / x ]_ [_ w / y ]_ C
25	20, 24	nfan 1545	. . . . . . 7 |- F/ y ( ( z e. A /\ w e. [_ z / x ]_ B ) /\ v = [_ z / x ]_ [_ w / y ]_ C )
26		eleq1 2203	. . . . . . . . . 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 2203	. . . . . . . . . 10 |- ( y = w -> ( y e. B <-> w e. B ) )
29		csbeq1a 3016	. . . . . . . . . . 11 |- ( x = z -> B = [_ z / x ]_ B )
30	29	eleq2d 2210	. . . . . . . . . 10 |- ( x = z -> ( w e. B <-> w e. [_ z / x ]_ B ) )
31	28, 30	sylan9bbr 459	. . . . . . . . 9 |- ( ( x = z /\ y = w ) -> ( y e. B <-> w e. [_ z / x ]_ B ) )
32	27, 31	anbi12d 465	. . . . . . . 8 |- ( ( x = z /\ y = w ) -> ( ( x e. A /\ y e. B ) <-> ( z e. A /\ w e. [_ z / x ]_ B ) ) )
33		csbeq1a 3016	. . . . . . . . . 10 |- ( y = w -> C = [_ w / y ]_ C )
34		csbeq1a 3016	. . . . . . . . . 10 |- ( x = z -> [_ w / y ]_ C = [_ z / x ]_ [_ w / y ]_ C )
35	33, 34	sylan9eqr 2195	. . . . . . . . 9 |- ( ( x = z /\ y = w ) -> C = [_ z / x ]_ [_ w / y ]_ C )
36	35	eqeq2d 2152	. . . . . . . 8 |- ( ( x = z /\ y = w ) -> ( v = C <-> v = [_ z / x ]_ [_ w / y ]_ C ) )
37	32, 36	anbi12d 465	. . . . . . 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 5853	. . . . . 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 5787	. . . . . 6 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , v >. | ( ( x e. A /\ y e. B ) /\ v = C ) }
40		df-mpo 5787	. . . . . 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 2171	. . . . 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 5870	. . . . 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 2171	. . . 4 |- F = ( v e. U_ z e. A ( { z } X. [_ z / x ]_ B ) |-> [_ ( 1st ` v ) / x ]_ [_ ( 2nd ` v ) / y ]_ C )
45	44	fmpt 5578	. . 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 1509	. . 3 |- F/ z A. y e. B C e. D
48	17	nfel1 2293	. . . 4 |- F/ x [_ z / x ]_ [_ w / y ]_ C e. D
49	14, 48	nfralxy 2474	. . 3 |- F/ x A. w e. [_ z / x ]_ B [_ z / x ]_ [_ w / y ]_ C e. D
50		nfv 1509	. . . . 5 |- F/ w C e. D
51	22	nfel1 2293	. . . . 5 |- F/ y [_ w / y ]_ C e. D
52	33	eleq1d 2209	. . . . 5 |- ( y = w -> ( C e. D <-> [_ w / y ]_ C e. D ) )
53	50, 51, 52	cbvral 2653	. . . 4 |- ( A. y e. B C e. D <-> A. w e. B [_ w / y ]_ C e. D )
54	34	eleq1d 2209	. . . . 5 |- ( x = z -> ( [_ w / y ]_ C e. D <-> [_ z / x ]_ [_ w / y ]_ C e. D ) )
55	29, 54	raleqbidv 2641	. . . 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 2653	. 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 2282	. . . 4 |- F/_ z ( { x } X. B )
59		nfcv 2282	. . . . 5 |- F/_ x { z }
60	59, 14	nfxp 4574	. . . 4 |- F/_ x ( { z } X. [_ z / x ]_ B )
61		sneq 3543	. . . . 5 |- ( x = z -> { x } = { z } )
62	61, 29	xpeq12d 4572	. . . 4 |- ( x = z -> ( { x } X. B ) = ( { z } X. [_ z / x ]_ B ) )
63	58, 60, 62	cbviun 3858	. . 3 |- U_ x e. A ( { x } X. B ) = U_ z e. A ( { z } X. [_ z / x ]_ B )
64	63	feq2i 5274	. 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 1332   e. wcel 1481   A.wral 2417   [_csb 3007   {csn 3532   <.cop 3535   U_ciun 3821   |-> cmpt 3997   X. cxp 4545   -->wf 5127   ` cfv 5131   {coprab 5783   e. cmpo 5784   1stc1st 6044   2ndc2nd 6045

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047

This theorem is referenced by:  fmpo  6107