MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fmpt2x Structured version   Unicode version

Theorem fmpt2x 6420
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.
StepHypRef Expression
1 vex 2961 . . . . . . . 8  |-  z  e. 
_V
2 vex 2961 . . . . . . . 8  |-  w  e. 
_V
31, 2op1std 6360 . . . . . . 7  |-  ( v  =  <. z ,  w >.  ->  ( 1st `  v
)  =  z )
43csbeq1d 3259 . . . . . 6  |-  ( v  =  <. z ,  w >.  ->  [_ ( 1st `  v
)  /  x ]_ [_ ( 2nd `  v
)  /  y ]_ C  =  [_ z  /  x ]_ [_ ( 2nd `  v )  /  y ]_ C )
51, 2op2ndd 6361 . . . . . . . 8  |-  ( v  =  <. z ,  w >.  ->  ( 2nd `  v
)  =  w )
65csbeq1d 3259 . . . . . . 7  |-  ( v  =  <. z ,  w >.  ->  [_ ( 2nd `  v
)  /  y ]_ C  =  [_ w  / 
y ]_ C )
76csbeq2dv 3278 . . . . . 6  |-  ( v  =  <. z ,  w >.  ->  [_ z  /  x ]_ [_ ( 2nd `  v
)  /  y ]_ C  =  [_ z  /  x ]_ [_ w  / 
y ]_ C )
84, 7eqtrd 2470 . . . . 5  |-  ( v  =  <. z ,  w >.  ->  [_ ( 1st `  v
)  /  x ]_ [_ ( 2nd `  v
)  /  y ]_ C  =  [_ z  /  x ]_ [_ w  / 
y ]_ C )
98eleq1d 2504 . . . 4  |-  ( v  =  <. z ,  w >.  ->  ( [_ ( 1st `  v )  /  x ]_ [_ ( 2nd `  v )  /  y ]_ C  e.  D  <->  [_ z  /  x ]_ [_ w  /  y ]_ C  e.  D )
)
109raliunxp 5017 . . 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 1630 . . . . . . 7  |-  F/ z ( ( x  e.  A  /\  y  e.  B )  /\  v  =  C )
12 nfv 1630 . . . . . . 7  |-  F/ w
( ( x  e.  A  /\  y  e.  B )  /\  v  =  C )
13 nfv 1630 . . . . . . . . 9  |-  F/ x  z  e.  A
14 nfcsb1v 3285 . . . . . . . . . 10  |-  F/_ x [_ z  /  x ]_ B
1514nfcri 2568 . . . . . . . . 9  |-  F/ x  w  e.  [_ z  /  x ]_ B
1613, 15nfan 1847 . . . . . . . 8  |-  F/ x
( z  e.  A  /\  w  e.  [_ z  /  x ]_ B )
17 nfcsb1v 3285 . . . . . . . . 9  |-  F/_ x [_ z  /  x ]_ [_ w  /  y ]_ C
1817nfeq2 2585 . . . . . . . 8  |-  F/ x  v  =  [_ z  /  x ]_ [_ w  / 
y ]_ C
1916, 18nfan 1847 . . . . . . 7  |-  F/ x
( ( z  e.  A  /\  w  e. 
[_ z  /  x ]_ B )  /\  v  =  [_ z  /  x ]_ [_ w  /  y ]_ C )
20 nfv 1630 . . . . . . . 8  |-  F/ y ( z  e.  A  /\  w  e.  [_ z  /  x ]_ B )
21 nfcv 2574 . . . . . . . . . 10  |-  F/_ y
z
22 nfcsb1v 3285 . . . . . . . . . 10  |-  F/_ y [_ w  /  y ]_ C
2321, 22nfcsb 3287 . . . . . . . . 9  |-  F/_ y [_ z  /  x ]_ [_ w  /  y ]_ C
2423nfeq2 2585 . . . . . . . 8  |-  F/ y  v  =  [_ z  /  x ]_ [_ w  /  y ]_ C
2520, 24nfan 1847 . . . . . . 7  |-  F/ y ( ( z  e.  A  /\  w  e. 
[_ z  /  x ]_ B )  /\  v  =  [_ z  /  x ]_ [_ w  /  y ]_ C )
26 eleq1 2498 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
2726adantr 453 . . . . . . . . 9  |-  ( ( x  =  z  /\  y  =  w )  ->  ( x  e.  A  <->  z  e.  A ) )
28 eleq1 2498 . . . . . . . . . 10  |-  ( y  =  w  ->  (
y  e.  B  <->  w  e.  B ) )
29 csbeq1a 3261 . . . . . . . . . . 11  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
3029eleq2d 2505 . . . . . . . . . 10  |-  ( x  =  z  ->  (
w  e.  B  <->  w  e.  [_ z  /  x ]_ B ) )
3128, 30sylan9bbr 683 . . . . . . . . 9  |-  ( ( x  =  z  /\  y  =  w )  ->  ( y  e.  B  <->  w  e.  [_ z  /  x ]_ B ) )
3227, 31anbi12d 693 . . . . . . . 8  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ( x  e.  A  /\  y  e.  B )  <->  ( z  e.  A  /\  w  e.  [_ z  /  x ]_ B ) ) )
33 csbeq1a 3261 . . . . . . . . . 10  |-  ( y  =  w  ->  C  =  [_ w  /  y ]_ C )
34 csbeq1a 3261 . . . . . . . . . 10  |-  ( x  =  z  ->  [_ w  /  y ]_ C  =  [_ z  /  x ]_ [_ w  /  y ]_ C )
3533, 34sylan9eqr 2492 . . . . . . . . 9  |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  [_ z  /  x ]_ [_ w  /  y ]_ C
)
3635eqeq2d 2449 . . . . . . . 8  |-  ( ( x  =  z  /\  y  =  w )  ->  ( v  =  C  <-> 
v  =  [_ z  /  x ]_ [_ w  /  y ]_ C
) )
3732, 36anbi12d 693 . . . . . . 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 ) ) )
3811, 12, 19, 25, 37cbvoprab12 6149 . . . . . 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 6089 . . . . . 6  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  v
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  v  =  C
) }
40 df-mpt2 6089 . . . . . 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
) }
4138, 39, 403eqtr4i 2468 . . . . 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 )
438mpt2mptx 6167 . . . . 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
)
4441, 42, 433eqtr4i 2468 . . . 4  |-  F  =  ( v  e.  U_ z  e.  A  ( { z }  X.  [_ z  /  x ]_ B )  |->  [_ ( 1st `  v )  /  x ]_ [_ ( 2nd `  v )  /  y ]_ C )
4544fmpt 5893 . . 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 )
4610, 45bitr3i 244 . 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 1630 . . 3  |-  F/ z A. y  e.  B  C  e.  D
4817nfel1 2584 . . . 4  |-  F/ x [_ z  /  x ]_ [_ w  /  y ]_ C  e.  D
4914, 48nfral 2761 . . 3  |-  F/ x A. w  e.  [_  z  /  x ]_ B [_ z  /  x ]_ [_ w  /  y ]_ C  e.  D
50 nfv 1630 . . . . 5  |-  F/ w  C  e.  D
5122nfel1 2584 . . . . 5  |-  F/ y
[_ w  /  y ]_ C  e.  D
5233eleq1d 2504 . . . . 5  |-  ( y  =  w  ->  ( C  e.  D  <->  [_ w  / 
y ]_ C  e.  D
) )
5350, 51, 52cbvral 2930 . . . 4  |-  ( A. y  e.  B  C  e.  D  <->  A. w  e.  B  [_ w  /  y ]_ C  e.  D )
5434eleq1d 2504 . . . . 5  |-  ( x  =  z  ->  ( [_ w  /  y ]_ C  e.  D  <->  [_ z  /  x ]_ [_ w  /  y ]_ C  e.  D )
)
5529, 54raleqbidv 2918 . . . 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
) )
5653, 55syl5bb 250 . . 3  |-  ( x  =  z  ->  ( A. y  e.  B  C  e.  D  <->  A. w  e.  [_  z  /  x ]_ B [_ z  /  x ]_ [_ w  / 
y ]_ C  e.  D
) )
5747, 49, 56cbvral 2930 . 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 2574 . . . 4  |-  F/_ z
( { x }  X.  B )
59 nfcv 2574 . . . . 5  |-  F/_ x { z }
6059, 14nfxp 4907 . . . 4  |-  F/_ x
( { z }  X.  [_ z  /  x ]_ B )
61 sneq 3827 . . . . 5  |-  ( x  =  z  ->  { x }  =  { z } )
6261, 29xpeq12d 4906 . . . 4  |-  ( x  =  z  ->  ( { x }  X.  B )  =  ( { z }  X.  [_ z  /  x ]_ B ) )
6358, 60, 62cbviun 4130 . . 3  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  U_ z  e.  A  ( {
z }  X.  [_ z  /  x ]_ B
)
6463feq2i 5589 . 2  |-  ( F : U_ x  e.  A  ( { x }  X.  B ) --> D  <-> 
F : U_ z  e.  A  ( {
z }  X.  [_ z  /  x ]_ B
) --> D )
6546, 57, 643bitr4i 270 1  |-  ( A. x  e.  A  A. y  e.  B  C  e.  D  <->  F : U_ x  e.  A  ( {
x }  X.  B
) --> D )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   [_csb 3253   {csn 3816   <.cop 3819   U_ciun 4095    e. cmpt 4269    X. cxp 4879   -->wf 5453   ` cfv 5457   {coprab 6085    e. cmpt2 6086   1stc1st 6350   2ndc2nd 6351
This theorem is referenced by:  fmpt2  6421  eldmcoa  14225  gsum2d2lem  15552  gsum2d2  15553  gsumcom2  15554  dmdprd  15564  dprdval  15566  dprd2d2  15607  ablfaclem2  15649  ptbasfi  17618  ptcmplem1  18088  prdsxmslem2  18564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353
  Copyright terms: Public domain W3C validator