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Theorem fmpt2x 5952
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 2622 . . . . . . . 8  |-  z  e. 
_V
2 vex 2622 . . . . . . . 8  |-  w  e. 
_V
31, 2op1std 5901 . . . . . . 7  |-  ( v  =  <. z ,  w >.  ->  ( 1st `  v
)  =  z )
43csbeq1d 2937 . . . . . 6  |-  ( v  =  <. z ,  w >.  ->  [_ ( 1st `  v
)  /  x ]_ [_ ( 2nd `  v
)  /  y ]_ C  =  [_ z  /  x ]_ [_ ( 2nd `  v )  /  y ]_ C )
51, 2op2ndd 5902 . . . . . . . 8  |-  ( v  =  <. z ,  w >.  ->  ( 2nd `  v
)  =  w )
65csbeq1d 2937 . . . . . . 7  |-  ( v  =  <. z ,  w >.  ->  [_ ( 2nd `  v
)  /  y ]_ C  =  [_ w  / 
y ]_ C )
76csbeq2dv 2954 . . . . . 6  |-  ( v  =  <. z ,  w >.  ->  [_ z  /  x ]_ [_ ( 2nd `  v
)  /  y ]_ C  =  [_ z  /  x ]_ [_ w  / 
y ]_ C )
84, 7eqtrd 2120 . . . . 5  |-  ( v  =  <. z ,  w >.  ->  [_ ( 1st `  v
)  /  x ]_ [_ ( 2nd `  v
)  /  y ]_ C  =  [_ z  /  x ]_ [_ w  / 
y ]_ C )
98eleq1d 2156 . . . 4  |-  ( v  =  <. z ,  w >.  ->  ( [_ ( 1st `  v )  /  x ]_ [_ ( 2nd `  v )  /  y ]_ C  e.  D  <->  [_ z  /  x ]_ [_ w  /  y ]_ C  e.  D )
)
109raliunxp 4565 . . 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 1466 . . . . . . 7  |-  F/ z ( ( x  e.  A  /\  y  e.  B )  /\  v  =  C )
12 nfv 1466 . . . . . . 7  |-  F/ w
( ( x  e.  A  /\  y  e.  B )  /\  v  =  C )
13 nfv 1466 . . . . . . . . 9  |-  F/ x  z  e.  A
14 nfcsb1v 2961 . . . . . . . . . 10  |-  F/_ x [_ z  /  x ]_ B
1514nfcri 2222 . . . . . . . . 9  |-  F/ x  w  e.  [_ z  /  x ]_ B
1613, 15nfan 1502 . . . . . . . 8  |-  F/ x
( z  e.  A  /\  w  e.  [_ z  /  x ]_ B )
17 nfcsb1v 2961 . . . . . . . . 9  |-  F/_ x [_ z  /  x ]_ [_ w  /  y ]_ C
1817nfeq2 2240 . . . . . . . 8  |-  F/ x  v  =  [_ z  /  x ]_ [_ w  / 
y ]_ C
1916, 18nfan 1502 . . . . . . 7  |-  F/ x
( ( z  e.  A  /\  w  e. 
[_ z  /  x ]_ B )  /\  v  =  [_ z  /  x ]_ [_ w  /  y ]_ C )
20 nfv 1466 . . . . . . . 8  |-  F/ y ( z  e.  A  /\  w  e.  [_ z  /  x ]_ B )
21 nfcv 2228 . . . . . . . . . 10  |-  F/_ y
z
22 nfcsb1v 2961 . . . . . . . . . 10  |-  F/_ y [_ w  /  y ]_ C
2321, 22nfcsb 2963 . . . . . . . . 9  |-  F/_ y [_ z  /  x ]_ [_ w  /  y ]_ C
2423nfeq2 2240 . . . . . . . 8  |-  F/ y  v  =  [_ z  /  x ]_ [_ w  /  y ]_ C
2520, 24nfan 1502 . . . . . . 7  |-  F/ y ( ( z  e.  A  /\  w  e. 
[_ z  /  x ]_ B )  /\  v  =  [_ z  /  x ]_ [_ w  /  y ]_ C )
26 eleq1 2150 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
2726adantr 270 . . . . . . . . 9  |-  ( ( x  =  z  /\  y  =  w )  ->  ( x  e.  A  <->  z  e.  A ) )
28 eleq1 2150 . . . . . . . . . 10  |-  ( y  =  w  ->  (
y  e.  B  <->  w  e.  B ) )
29 csbeq1a 2939 . . . . . . . . . . 11  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
3029eleq2d 2157 . . . . . . . . . 10  |-  ( x  =  z  ->  (
w  e.  B  <->  w  e.  [_ z  /  x ]_ B ) )
3128, 30sylan9bbr 451 . . . . . . . . 9  |-  ( ( x  =  z  /\  y  =  w )  ->  ( y  e.  B  <->  w  e.  [_ z  /  x ]_ B ) )
3227, 31anbi12d 457 . . . . . . . 8  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ( x  e.  A  /\  y  e.  B )  <->  ( z  e.  A  /\  w  e.  [_ z  /  x ]_ B ) ) )
33 csbeq1a 2939 . . . . . . . . . 10  |-  ( y  =  w  ->  C  =  [_ w  /  y ]_ C )
34 csbeq1a 2939 . . . . . . . . . 10  |-  ( x  =  z  ->  [_ w  /  y ]_ C  =  [_ z  /  x ]_ [_ w  /  y ]_ C )
3533, 34sylan9eqr 2142 . . . . . . . . 9  |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  [_ z  /  x ]_ [_ w  /  y ]_ C
)
3635eqeq2d 2099 . . . . . . . 8  |-  ( ( x  =  z  /\  y  =  w )  ->  ( v  =  C  <-> 
v  =  [_ z  /  x ]_ [_ w  /  y ]_ C
) )
3732, 36anbi12d 457 . . . . . . 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 5704 . . . . . 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 5639 . . . . . 6  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  v
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  v  =  C
) }
40 df-mpt2 5639 . . . . . 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 2118 . . . . 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 5721 . . . . 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 2118 . . . 4  |-  F  =  ( v  e.  U_ z  e.  A  ( { z }  X.  [_ z  /  x ]_ B )  |->  [_ ( 1st `  v )  /  x ]_ [_ ( 2nd `  v )  /  y ]_ C )
4544fmpt 5433 . . 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 184 . 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 1466 . . 3  |-  F/ z A. y  e.  B  C  e.  D
4817nfel1 2239 . . . 4  |-  F/ x [_ z  /  x ]_ [_ w  /  y ]_ C  e.  D
4914, 48nfralxy 2414 . . 3  |-  F/ x A. w  e.  [_  z  /  x ]_ B [_ z  /  x ]_ [_ w  /  y ]_ C  e.  D
50 nfv 1466 . . . . 5  |-  F/ w  C  e.  D
5122nfel1 2239 . . . . 5  |-  F/ y
[_ w  /  y ]_ C  e.  D
5233eleq1d 2156 . . . . 5  |-  ( y  =  w  ->  ( C  e.  D  <->  [_ w  / 
y ]_ C  e.  D
) )
5350, 51, 52cbvral 2586 . . . 4  |-  ( A. y  e.  B  C  e.  D  <->  A. w  e.  B  [_ w  /  y ]_ C  e.  D )
5434eleq1d 2156 . . . . 5  |-  ( x  =  z  ->  ( [_ w  /  y ]_ C  e.  D  <->  [_ z  /  x ]_ [_ w  /  y ]_ C  e.  D )
)
5529, 54raleqbidv 2574 . . . 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 190 . . 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 2586 . 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 2228 . . . 4  |-  F/_ z
( { x }  X.  B )
59 nfcv 2228 . . . . 5  |-  F/_ x { z }
6059, 14nfxp 4454 . . . 4  |-  F/_ x
( { z }  X.  [_ z  /  x ]_ B )
61 sneq 3452 . . . . 5  |-  ( x  =  z  ->  { x }  =  { z } )
6261, 29xpeq12d 4453 . . . 4  |-  ( x  =  z  ->  ( { x }  X.  B )  =  ( { z }  X.  [_ z  /  x ]_ B ) )
6358, 60, 62cbviun 3762 . . 3  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  U_ z  e.  A  ( {
z }  X.  [_ z  /  x ]_ B
)
6463feq2i 5141 . 2  |-  ( F : U_ x  e.  A  ( { x }  X.  B ) --> D  <-> 
F : U_ z  e.  A  ( {
z }  X.  [_ z  /  x ]_ B
) --> D )
6546, 57, 643bitr4i 210 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:    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   A.wral 2359   [_csb 2931   {csn 3441   <.cop 3444   U_ciun 3725    |-> cmpt 3891    X. cxp 4426   -->wf 4998   ` cfv 5002   {coprab 5635    |-> cmpt2 5636   1stc1st 5891   2ndc2nd 5892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894
This theorem is referenced by:  fmpt2  5953
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