MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmmpt2ssx Structured version   Unicode version

Theorem dmmpt2ssx 6838
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 2616 . . . . 5  |-  F/_ u B
2 nfcsb1v 3436 . . . . 5  |-  F/_ x [_ u  /  x ]_ B
3 nfcv 2616 . . . . 5  |-  F/_ u C
4 nfcv 2616 . . . . 5  |-  F/_ v C
5 nfcsb1v 3436 . . . . 5  |-  F/_ x [_ u  /  x ]_ [_ v  /  y ]_ C
6 nfcv 2616 . . . . . 6  |-  F/_ y
u
7 nfcsb1v 3436 . . . . . 6  |-  F/_ y [_ v  /  y ]_ C
86, 7nfcsb 3438 . . . . 5  |-  F/_ y [_ u  /  x ]_ [_ v  /  y ]_ C
9 csbeq1a 3429 . . . . 5  |-  ( x  =  u  ->  B  =  [_ u  /  x ]_ B )
10 csbeq1a 3429 . . . . . 6  |-  ( y  =  v  ->  C  =  [_ v  /  y ]_ C )
11 csbeq1a 3429 . . . . . 6  |-  ( x  =  u  ->  [_ v  /  y ]_ C  =  [_ u  /  x ]_ [_ v  /  y ]_ C )
1210, 11sylan9eqr 2517 . . . . 5  |-  ( ( x  =  u  /\  y  =  v )  ->  C  =  [_ u  /  x ]_ [_ v  /  y ]_ C
)
131, 2, 3, 4, 5, 8, 9, 12cbvmpt2x 6348 . . . 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 3109 . . . . . . . 8  |-  u  e. 
_V
16 vex 3109 . . . . . . . 8  |-  v  e. 
_V
1715, 16op1std 6783 . . . . . . 7  |-  ( t  =  <. u ,  v
>.  ->  ( 1st `  t
)  =  u )
1817csbeq1d 3427 . . . . . 6  |-  ( t  =  <. u ,  v
>.  ->  [_ ( 1st `  t
)  /  x ]_ [_ ( 2nd `  t
)  /  y ]_ C  =  [_ u  /  x ]_ [_ ( 2nd `  t )  /  y ]_ C )
1915, 16op2ndd 6784 . . . . . . . 8  |-  ( t  =  <. u ,  v
>.  ->  ( 2nd `  t
)  =  v )
2019csbeq1d 3427 . . . . . . 7  |-  ( t  =  <. u ,  v
>.  ->  [_ ( 2nd `  t
)  /  y ]_ C  =  [_ v  / 
y ]_ C )
2120csbeq2dv 3831 . . . . . 6  |-  ( t  =  <. u ,  v
>.  ->  [_ u  /  x ]_ [_ ( 2nd `  t
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
2218, 21eqtrd 2495 . . . . 5  |-  ( t  =  <. u ,  v
>.  ->  [_ ( 1st `  t
)  /  x ]_ [_ ( 2nd `  t
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
2322mpt2mptx 6366 . . . 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 2493 . . 3  |-  F  =  ( t  e.  U_ u  e.  A  ( { u }  X.  [_ u  /  x ]_ B )  |->  [_ ( 1st `  t )  /  x ]_ [_ ( 2nd `  t )  /  y ]_ C )
2524dmmptss 5486 . 2  |-  dom  F  C_ 
U_ u  e.  A  ( { u }  X.  [_ u  /  x ]_ B )
26 nfcv 2616 . . 3  |-  F/_ u
( { x }  X.  B )
27 nfcv 2616 . . . 4  |-  F/_ x { u }
2827, 2nfxp 5015 . . 3  |-  F/_ x
( { u }  X.  [_ u  /  x ]_ B )
29 sneq 4026 . . . 4  |-  ( x  =  u  ->  { x }  =  { u } )
3029, 9xpeq12d 5013 . . 3  |-  ( x  =  u  ->  ( { x }  X.  B )  =  ( { u }  X.  [_ u  /  x ]_ B ) )
3126, 28, 30cbviun 4352 . 2  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  U_ u  e.  A  ( {
u }  X.  [_ u  /  x ]_ B
)
3225, 31sseqtr4i 3522 1  |-  dom  F  C_ 
U_ x  e.  A  ( { x }  X.  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398   [_csb 3420    C_ wss 3461   {csn 4016   <.cop 4022   U_ciun 4315    |-> cmpt 4497    X. cxp 4986   dom cdm 4988   ` cfv 5570    |-> cmpt2 6272   1stc1st 6771   2ndc2nd 6772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fv 5578  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774
This theorem is referenced by:  mpt2exxg  6847  mpt2xopn0yelv  6933  mpt2xopxnop0  6935  dmcoass  15544  ply1frcl  18550  dvbsss  22472  perfdvf  22473
  Copyright terms: Public domain W3C validator