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Theorem dmmpossx 6254
Description: The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)
Hypothesis
Ref Expression
fmpox.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
dmmpossx  |-  dom  F  C_ 
U_ x  e.  A  ( { x }  X.  B )
Distinct variable groups:    x, y, A   
y, B
Allowed substitution hints:    B( x)    C( x, y)    F( x, y)

Proof of Theorem dmmpossx
Dummy variables  u  t  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2336 . . . . 5  |-  F/_ u B
2 nfcsb1v 3114 . . . . 5  |-  F/_ x [_ u  /  x ]_ B
3 nfcv 2336 . . . . 5  |-  F/_ u C
4 nfcv 2336 . . . . 5  |-  F/_ v C
5 nfcsb1v 3114 . . . . 5  |-  F/_ x [_ u  /  x ]_ [_ v  /  y ]_ C
6 nfcv 2336 . . . . . 6  |-  F/_ y
u
7 nfcsb1v 3114 . . . . . 6  |-  F/_ y [_ v  /  y ]_ C
86, 7nfcsb 3119 . . . . 5  |-  F/_ y [_ u  /  x ]_ [_ v  /  y ]_ C
9 csbeq1a 3090 . . . . 5  |-  ( x  =  u  ->  B  =  [_ u  /  x ]_ B )
10 csbeq1a 3090 . . . . . 6  |-  ( y  =  v  ->  C  =  [_ v  /  y ]_ C )
11 csbeq1a 3090 . . . . . 6  |-  ( x  =  u  ->  [_ v  /  y ]_ C  =  [_ u  /  x ]_ [_ v  /  y ]_ C )
1210, 11sylan9eqr 2248 . . . . 5  |-  ( ( x  =  u  /\  y  =  v )  ->  C  =  [_ u  /  x ]_ [_ v  /  y ]_ C
)
131, 2, 3, 4, 5, 8, 9, 12cbvmpox 5997 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( u  e.  A ,  v  e. 
[_ u  /  x ]_ B  |->  [_ u  /  x ]_ [_ v  /  y ]_ C
)
14 fmpox.1 . . . 4  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
15 vex 2763 . . . . . . . 8  |-  u  e. 
_V
16 vex 2763 . . . . . . . 8  |-  v  e. 
_V
1715, 16op1std 6203 . . . . . . 7  |-  ( t  =  <. u ,  v
>.  ->  ( 1st `  t
)  =  u )
1817csbeq1d 3088 . . . . . 6  |-  ( t  =  <. u ,  v
>.  ->  [_ ( 1st `  t
)  /  x ]_ [_ ( 2nd `  t
)  /  y ]_ C  =  [_ u  /  x ]_ [_ ( 2nd `  t )  /  y ]_ C )
1915, 16op2ndd 6204 . . . . . . . 8  |-  ( t  =  <. u ,  v
>.  ->  ( 2nd `  t
)  =  v )
2019csbeq1d 3088 . . . . . . 7  |-  ( t  =  <. u ,  v
>.  ->  [_ ( 2nd `  t
)  /  y ]_ C  =  [_ v  / 
y ]_ C )
2120csbeq2dv 3107 . . . . . 6  |-  ( t  =  <. u ,  v
>.  ->  [_ u  /  x ]_ [_ ( 2nd `  t
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
2218, 21eqtrd 2226 . . . . 5  |-  ( t  =  <. u ,  v
>.  ->  [_ ( 1st `  t
)  /  x ]_ [_ ( 2nd `  t
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
2322mpomptx 6010 . . . 4  |-  ( t  e.  U_ u  e.  A  ( { u }  X.  [_ u  /  x ]_ B )  |->  [_ ( 1st `  t )  /  x ]_ [_ ( 2nd `  t )  / 
y ]_ C )  =  ( u  e.  A ,  v  e.  [_ u  /  x ]_ B  |->  [_ u  /  x ]_ [_ v  /  y ]_ C
)
2413, 14, 233eqtr4i 2224 . . 3  |-  F  =  ( t  e.  U_ u  e.  A  ( { u }  X.  [_ u  /  x ]_ B )  |->  [_ ( 1st `  t )  /  x ]_ [_ ( 2nd `  t )  /  y ]_ C )
2524dmmptss 5163 . 2  |-  dom  F  C_ 
U_ u  e.  A  ( { u }  X.  [_ u  /  x ]_ B )
26 nfcv 2336 . . 3  |-  F/_ u
( { x }  X.  B )
27 nfcv 2336 . . . 4  |-  F/_ x { u }
2827, 2nfxp 4687 . . 3  |-  F/_ x
( { u }  X.  [_ u  /  x ]_ B )
29 sneq 3630 . . . 4  |-  ( x  =  u  ->  { x }  =  { u } )
3029, 9xpeq12d 4685 . . 3  |-  ( x  =  u  ->  ( { x }  X.  B )  =  ( { u }  X.  [_ u  /  x ]_ B ) )
3126, 28, 30cbviun 3950 . 2  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  U_ u  e.  A  ( {
u }  X.  [_ u  /  x ]_ B
)
3225, 31sseqtrri 3215 1  |-  dom  F  C_ 
U_ x  e.  A  ( { x }  X.  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1364   [_csb 3081    C_ wss 3154   {csn 3619   <.cop 3622   U_ciun 3913    |-> cmpt 4091    X. cxp 4658   dom cdm 4660   ` cfv 5255    e. cmpo 5921   1stc1st 6193   2ndc2nd 6194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fv 5263  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196
This theorem is referenced by:  mpoexxg  6265  mpoxopn0yelv  6294
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