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Theorem dmmpt2ssx2 40626
Description: The domain of a mapping is a subset of its base classes expressed as union of Cartesian products over its second component, analogous to dmmpt2ssx 6877. (Contributed by AV, 30-Mar-2019.)
Hypothesis
Ref Expression
dmmpt2ssx2.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
dmmpt2ssx2  |-  dom  F  C_ 
U_ y  e.  B  ( A  X.  { y } )
Distinct variable groups:    x, A    x, y, B
Allowed substitution hints:    A( y)    C( x, y)    F( x, y)

Proof of Theorem dmmpt2ssx2
Dummy variables  u  t  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2612 . . . . 5  |-  F/_ u A
2 nfcsb1v 3365 . . . . 5  |-  F/_ y [_ u  /  y ]_ A
3 nfcv 2612 . . . . 5  |-  F/_ u C
4 nfcv 2612 . . . . 5  |-  F/_ v C
5 nfcv 2612 . . . . . 6  |-  F/_ x u
6 nfcsb1v 3365 . . . . . 6  |-  F/_ x [_ v  /  x ]_ C
75, 6nfcsb 3367 . . . . 5  |-  F/_ x [_ u  /  y ]_ [_ v  /  x ]_ C
8 nfcsb1v 3365 . . . . 5  |-  F/_ y [_ u  /  y ]_ [_ v  /  x ]_ C
9 csbeq1a 3358 . . . . 5  |-  ( y  =  u  ->  A  =  [_ u  /  y ]_ A )
10 csbeq1a 3358 . . . . . 6  |-  ( x  =  v  ->  C  =  [_ v  /  x ]_ C )
11 csbeq1a 3358 . . . . . 6  |-  ( y  =  u  ->  [_ v  /  x ]_ C  = 
[_ u  /  y ]_ [_ v  /  x ]_ C )
1210, 11sylan9eqr 2527 . . . . 5  |-  ( ( y  =  u  /\  x  =  v )  ->  C  =  [_ u  /  y ]_ [_ v  /  x ]_ C )
131, 2, 3, 4, 7, 8, 9, 12cbvmpt2x2 40625 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( v  e. 
[_ u  /  y ]_ A ,  u  e.  B  |->  [_ u  /  y ]_ [_ v  /  x ]_ C )
14 dmmpt2ssx2.1 . . . 4  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
15 vex 3034 . . . . . . . 8  |-  v  e. 
_V
16 vex 3034 . . . . . . . 8  |-  u  e. 
_V
1715, 16op2ndd 6823 . . . . . . 7  |-  ( t  =  <. v ,  u >.  ->  ( 2nd `  t
)  =  u )
1817csbeq1d 3356 . . . . . 6  |-  ( t  =  <. v ,  u >.  ->  [_ ( 2nd `  t
)  /  y ]_ [_ ( 1st `  t
)  /  x ]_ C  =  [_ u  / 
y ]_ [_ ( 1st `  t )  /  x ]_ C )
1915, 16op1std 6822 . . . . . . . 8  |-  ( t  =  <. v ,  u >.  ->  ( 1st `  t
)  =  v )
2019csbeq1d 3356 . . . . . . 7  |-  ( t  =  <. v ,  u >.  ->  [_ ( 1st `  t
)  /  x ]_ C  =  [_ v  /  x ]_ C )
2120csbeq2dv 3785 . . . . . 6  |-  ( t  =  <. v ,  u >.  ->  [_ u  /  y ]_ [_ ( 1st `  t
)  /  x ]_ C  =  [_ u  / 
y ]_ [_ v  /  x ]_ C )
2218, 21eqtrd 2505 . . . . 5  |-  ( t  =  <. v ,  u >.  ->  [_ ( 2nd `  t
)  /  y ]_ [_ ( 1st `  t
)  /  x ]_ C  =  [_ u  / 
y ]_ [_ v  /  x ]_ C )
2322mpt2mptx2 40624 . . . 4  |-  ( t  e.  U_ u  e.  B  ( [_ u  /  y ]_ A  X.  { u } ) 
|->  [_ ( 2nd `  t
)  /  y ]_ [_ ( 1st `  t
)  /  x ]_ C )  =  ( v  e.  [_ u  /  y ]_ A ,  u  e.  B  |-> 
[_ u  /  y ]_ [_ v  /  x ]_ C )
2413, 14, 233eqtr4i 2503 . . 3  |-  F  =  ( t  e.  U_ u  e.  B  ( [_ u  /  y ]_ A  X.  { u } )  |->  [_ ( 2nd `  t )  / 
y ]_ [_ ( 1st `  t )  /  x ]_ C )
2524dmmptss 5338 . 2  |-  dom  F  C_ 
U_ u  e.  B  ( [_ u  /  y ]_ A  X.  { u } )
26 nfcv 2612 . . 3  |-  F/_ u
( A  X.  {
y } )
27 nfcv 2612 . . . 4  |-  F/_ y { u }
282, 27nfxp 4866 . . 3  |-  F/_ y
( [_ u  /  y ]_ A  X.  { u } )
29 sneq 3969 . . . 4  |-  ( y  =  u  ->  { y }  =  { u } )
309, 29xpeq12d 4864 . . 3  |-  ( y  =  u  ->  ( A  X.  { y } )  =  ( [_ u  /  y ]_ A  X.  { u } ) )
3126, 28, 30cbviun 4306 . 2  |-  U_ y  e.  B  ( A  X.  { y } )  =  U_ u  e.  B  ( [_ u  /  y ]_ A  X.  { u } )
3225, 31sseqtr4i 3451 1  |-  dom  F  C_ 
U_ y  e.  B  ( A  X.  { y } )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452   [_csb 3349    C_ wss 3390   {csn 3959   <.cop 3965   U_ciun 4269    |-> cmpt 4454    X. cxp 4837   dom cdm 4839   ` cfv 5589    |-> cmpt2 6310   1stc1st 6810   2ndc2nd 6811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813
This theorem is referenced by:  mpt2exxg2  40627
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