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Theorem dmmpt2ssx 5825
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 2178 . . . . 5  |-  F/_ u B
2 nfcsb1v 2882 . . . . 5  |-  F/_ x [_ u  /  x ]_ B
3 nfcv 2178 . . . . 5  |-  F/_ u C
4 nfcv 2178 . . . . 5  |-  F/_ v C
5 nfcsb1v 2882 . . . . 5  |-  F/_ x [_ u  /  x ]_ [_ v  /  y ]_ C
6 nfcv 2178 . . . . . 6  |-  F/_ y
u
7 nfcsb1v 2882 . . . . . 6  |-  F/_ y [_ v  /  y ]_ C
86, 7nfcsb 2884 . . . . 5  |-  F/_ y [_ u  /  x ]_ [_ v  /  y ]_ C
9 csbeq1a 2860 . . . . 5  |-  ( x  =  u  ->  B  =  [_ u  /  x ]_ B )
10 csbeq1a 2860 . . . . . 6  |-  ( y  =  v  ->  C  =  [_ v  /  y ]_ C )
11 csbeq1a 2860 . . . . . 6  |-  ( x  =  u  ->  [_ v  /  y ]_ C  =  [_ u  /  x ]_ [_ v  /  y ]_ C )
1210, 11sylan9eqr 2094 . . . . 5  |-  ( ( x  =  u  /\  y  =  v )  ->  C  =  [_ u  /  x ]_ [_ v  /  y ]_ C
)
131, 2, 3, 4, 5, 8, 9, 12cbvmpt2x 5582 . . . 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 2560 . . . . . . . 8  |-  u  e. 
_V
16 vex 2560 . . . . . . . 8  |-  v  e. 
_V
1715, 16op1std 5775 . . . . . . 7  |-  ( t  =  <. u ,  v
>.  ->  ( 1st `  t
)  =  u )
1817csbeq1d 2858 . . . . . 6  |-  ( t  =  <. u ,  v
>.  ->  [_ ( 1st `  t
)  /  x ]_ [_ ( 2nd `  t
)  /  y ]_ C  =  [_ u  /  x ]_ [_ ( 2nd `  t )  /  y ]_ C )
1915, 16op2ndd 5776 . . . . . . . 8  |-  ( t  =  <. u ,  v
>.  ->  ( 2nd `  t
)  =  v )
2019csbeq1d 2858 . . . . . . 7  |-  ( t  =  <. u ,  v
>.  ->  [_ ( 2nd `  t
)  /  y ]_ C  =  [_ v  / 
y ]_ C )
2120csbeq2dv 2875 . . . . . 6  |-  ( t  =  <. u ,  v
>.  ->  [_ u  /  x ]_ [_ ( 2nd `  t
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
2218, 21eqtrd 2072 . . . . 5  |-  ( t  =  <. u ,  v
>.  ->  [_ ( 1st `  t
)  /  x ]_ [_ ( 2nd `  t
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
2322mpt2mptx 5595 . . . 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 2070 . . 3  |-  F  =  ( t  e.  U_ u  e.  A  ( { u }  X.  [_ u  /  x ]_ B )  |->  [_ ( 1st `  t )  /  x ]_ [_ ( 2nd `  t )  /  y ]_ C )
2524dmmptss 4817 . 2  |-  dom  F  C_ 
U_ u  e.  A  ( { u }  X.  [_ u  /  x ]_ B )
26 nfcv 2178 . . 3  |-  F/_ u
( { x }  X.  B )
27 nfcv 2178 . . . 4  |-  F/_ x { u }
2827, 2nfxp 4371 . . 3  |-  F/_ x
( { u }  X.  [_ u  /  x ]_ B )
29 sneq 3386 . . . 4  |-  ( x  =  u  ->  { x }  =  { u } )
3029, 9xpeq12d 4370 . . 3  |-  ( x  =  u  ->  ( { x }  X.  B )  =  ( { u }  X.  [_ u  /  x ]_ B ) )
3126, 28, 30cbviun 3694 . 2  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  U_ u  e.  A  ( {
u }  X.  [_ u  /  x ]_ B
)
3225, 31sseqtr4i 2978 1  |-  dom  F  C_ 
U_ x  e.  A  ( { x }  X.  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1243   [_csb 2852    C_ wss 2917   {csn 3375   <.cop 3378   U_ciun 3657    |-> cmpt 3818    X. cxp 4343   dom cdm 4345   ` cfv 4902    |-> cmpt2 5514   1stc1st 5765   2ndc2nd 5766
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fv 4910  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768
This theorem is referenced by:  mpt2exxg  5833  mpt2xopn0yelv  5854
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