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Theorem dmmpt2ssx 6744
Description: The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)
Hypothesis
Ref Expression
fmpt2x.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
dmmpt2ssx  |-  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 dmmpt2ssx
Dummy variables  u  t  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2614 . . . . 5  |-  F/_ u B
2 nfcsb1v 3406 . . . . 5  |-  F/_ x [_ u  /  x ]_ B
3 nfcv 2614 . . . . 5  |-  F/_ u C
4 nfcv 2614 . . . . 5  |-  F/_ v C
5 nfcsb1v 3406 . . . . 5  |-  F/_ x [_ u  /  x ]_ [_ v  /  y ]_ C
6 nfcv 2614 . . . . . 6  |-  F/_ y
u
7 nfcsb1v 3406 . . . . . 6  |-  F/_ y [_ v  /  y ]_ C
86, 7nfcsb 3408 . . . . 5  |-  F/_ y [_ u  /  x ]_ [_ v  /  y ]_ C
9 csbeq1a 3399 . . . . 5  |-  ( x  =  u  ->  B  =  [_ u  /  x ]_ B )
10 csbeq1a 3399 . . . . . 6  |-  ( y  =  v  ->  C  =  [_ v  /  y ]_ C )
11 csbeq1a 3399 . . . . . 6  |-  ( x  =  u  ->  [_ v  /  y ]_ C  =  [_ u  /  x ]_ [_ v  /  y ]_ C )
1210, 11sylan9eqr 2515 . . . . 5  |-  ( ( x  =  u  /\  y  =  v )  ->  C  =  [_ u  /  x ]_ [_ v  /  y ]_ C
)
131, 2, 3, 4, 5, 8, 9, 12cbvmpt2x 6268 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( u  e.  A ,  v  e. 
[_ u  /  x ]_ B  |->  [_ u  /  x ]_ [_ v  /  y ]_ C
)
14 fmpt2x.1 . . . 4  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
15 vex 3075 . . . . . . . 8  |-  u  e. 
_V
16 vex 3075 . . . . . . . 8  |-  v  e. 
_V
1715, 16op1std 6692 . . . . . . 7  |-  ( t  =  <. u ,  v
>.  ->  ( 1st `  t
)  =  u )
1817csbeq1d 3397 . . . . . 6  |-  ( t  =  <. u ,  v
>.  ->  [_ ( 1st `  t
)  /  x ]_ [_ ( 2nd `  t
)  /  y ]_ C  =  [_ u  /  x ]_ [_ ( 2nd `  t )  /  y ]_ C )
1915, 16op2ndd 6693 . . . . . . . 8  |-  ( t  =  <. u ,  v
>.  ->  ( 2nd `  t
)  =  v )
2019csbeq1d 3397 . . . . . . 7  |-  ( t  =  <. u ,  v
>.  ->  [_ ( 2nd `  t
)  /  y ]_ C  =  [_ v  / 
y ]_ C )
2120csbeq2dv 3790 . . . . . 6  |-  ( t  =  <. u ,  v
>.  ->  [_ u  /  x ]_ [_ ( 2nd `  t
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
2218, 21eqtrd 2493 . . . . 5  |-  ( t  =  <. u ,  v
>.  ->  [_ ( 1st `  t
)  /  x ]_ [_ ( 2nd `  t
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
2322mpt2mptx 6286 . . . 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 2491 . . 3  |-  F  =  ( t  e.  U_ u  e.  A  ( { u }  X.  [_ u  /  x ]_ B )  |->  [_ ( 1st `  t )  /  x ]_ [_ ( 2nd `  t )  /  y ]_ C )
2524dmmptss 5437 . 2  |-  dom  F  C_ 
U_ u  e.  A  ( { u }  X.  [_ u  /  x ]_ B )
26 nfcv 2614 . . 3  |-  F/_ u
( { x }  X.  B )
27 nfcv 2614 . . . 4  |-  F/_ x { u }
2827, 2nfxp 4969 . . 3  |-  F/_ x
( { u }  X.  [_ u  /  x ]_ B )
29 sneq 3990 . . . 4  |-  ( x  =  u  ->  { x }  =  { u } )
3029, 9xpeq12d 4968 . . 3  |-  ( x  =  u  ->  ( { x }  X.  B )  =  ( { u }  X.  [_ u  /  x ]_ B ) )
3126, 28, 30cbviun 4310 . 2  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  U_ u  e.  A  ( {
u }  X.  [_ u  /  x ]_ B
)
3225, 31sseqtr4i 3492 1  |-  dom  F  C_ 
U_ x  e.  A  ( { x }  X.  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   [_csb 3390    C_ wss 3431   {csn 3980   <.cop 3986   U_ciun 4274    |-> cmpt 4453    X. cxp 4941   dom cdm 4943   ` cfv 5521    |-> cmpt2 6197   1stc1st 6680   2ndc2nd 6681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fv 5529  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683
This theorem is referenced by:  mpt2exxg  6752  mpt2xopn0yelv  6835  mpt2xopxnop0  6837  dmcoass  15048  ply1frcl  17873  dvbsss  21505  perfdvf  21506
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