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Theorem dmmpt2ssx2 30876
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 6750. (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 2616 . . . . 5  |-  F/_ u A
2 nfcsb1v 3412 . . . . 5  |-  F/_ y [_ u  /  y ]_ A
3 nfcv 2616 . . . . 5  |-  F/_ u C
4 nfcv 2616 . . . . 5  |-  F/_ v C
5 nfcv 2616 . . . . . 6  |-  F/_ x u
6 nfcsb1v 3412 . . . . . 6  |-  F/_ x [_ v  /  x ]_ C
75, 6nfcsb 3414 . . . . 5  |-  F/_ x [_ u  /  y ]_ [_ v  /  x ]_ C
8 nfcsb1v 3412 . . . . 5  |-  F/_ y [_ u  /  y ]_ [_ v  /  x ]_ C
9 csbeq1a 3405 . . . . 5  |-  ( y  =  u  ->  A  =  [_ u  /  y ]_ A )
10 csbeq1a 3405 . . . . . 6  |-  ( x  =  v  ->  C  =  [_ v  /  x ]_ C )
11 csbeq1a 3405 . . . . . 6  |-  ( y  =  u  ->  [_ v  /  x ]_ C  = 
[_ u  /  y ]_ [_ v  /  x ]_ C )
1210, 11sylan9eqr 2517 . . . . 5  |-  ( ( y  =  u  /\  x  =  v )  ->  C  =  [_ u  /  y ]_ [_ v  /  x ]_ C )
131, 2, 3, 4, 7, 8, 9, 12cbvmpt2x2 30875 . . . 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 3081 . . . . . . . 8  |-  v  e. 
_V
16 vex 3081 . . . . . . . 8  |-  u  e. 
_V
1715, 16op2ndd 6699 . . . . . . 7  |-  ( t  =  <. v ,  u >.  ->  ( 2nd `  t
)  =  u )
1817csbeq1d 3403 . . . . . 6  |-  ( t  =  <. v ,  u >.  ->  [_ ( 2nd `  t
)  /  y ]_ [_ ( 1st `  t
)  /  x ]_ C  =  [_ u  / 
y ]_ [_ ( 1st `  t )  /  x ]_ C )
1915, 16op1std 6698 . . . . . . . 8  |-  ( t  =  <. v ,  u >.  ->  ( 1st `  t
)  =  v )
2019csbeq1d 3403 . . . . . . 7  |-  ( t  =  <. v ,  u >.  ->  [_ ( 1st `  t
)  /  x ]_ C  =  [_ v  /  x ]_ C )
2120csbeq2dv 3796 . . . . . 6  |-  ( t  =  <. v ,  u >.  ->  [_ u  /  y ]_ [_ ( 1st `  t
)  /  x ]_ C  =  [_ u  / 
y ]_ [_ v  /  x ]_ C )
2218, 21eqtrd 2495 . . . . 5  |-  ( t  =  <. v ,  u >.  ->  [_ ( 2nd `  t
)  /  y ]_ [_ ( 1st `  t
)  /  x ]_ C  =  [_ u  / 
y ]_ [_ v  /  x ]_ C )
2322mpt2mptx2 30873 . . . 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 2493 . . 3  |-  F  =  ( t  e.  U_ u  e.  B  ( [_ u  /  y ]_ A  X.  { u } )  |->  [_ ( 2nd `  t )  / 
y ]_ [_ ( 1st `  t )  /  x ]_ C )
2524dmmptss 5443 . 2  |-  dom  F  C_ 
U_ u  e.  B  ( [_ u  /  y ]_ A  X.  { u } )
26 nfcv 2616 . . 3  |-  F/_ u
( A  X.  {
y } )
27 nfcv 2616 . . . 4  |-  F/_ y { u }
282, 27nfxp 4975 . . 3  |-  F/_ y
( [_ u  /  y ]_ A  X.  { u } )
29 sneq 3996 . . . 4  |-  ( y  =  u  ->  { y }  =  { u } )
309, 29xpeq12d 4974 . . 3  |-  ( y  =  u  ->  ( A  X.  { y } )  =  ( [_ u  /  y ]_ A  X.  { u } ) )
3126, 28, 30cbviun 4316 . 2  |-  U_ y  e.  B  ( A  X.  { y } )  =  U_ u  e.  B  ( [_ u  /  y ]_ A  X.  { u } )
3225, 31sseqtr4i 3498 1  |-  dom  F  C_ 
U_ y  e.  B  ( A  X.  { y } )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   [_csb 3396    C_ wss 3437   {csn 3986   <.cop 3992   U_ciun 4280    |-> cmpt 4459    X. cxp 4947   dom cdm 4949   ` cfv 5527    |-> cmpt2 6203   1stc1st 6686   2ndc2nd 6687
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 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fv 5535  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689
This theorem is referenced by:  mpt2exxg2  30877
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