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Theorem dmmpt2ssx2 33070
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 6864. (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 2619 . . . . 5  |-  F/_ u A
2 nfcsb1v 3446 . . . . 5  |-  F/_ y [_ u  /  y ]_ A
3 nfcv 2619 . . . . 5  |-  F/_ u C
4 nfcv 2619 . . . . 5  |-  F/_ v C
5 nfcv 2619 . . . . . 6  |-  F/_ x u
6 nfcsb1v 3446 . . . . . 6  |-  F/_ x [_ v  /  x ]_ C
75, 6nfcsb 3448 . . . . 5  |-  F/_ x [_ u  /  y ]_ [_ v  /  x ]_ C
8 nfcsb1v 3446 . . . . 5  |-  F/_ y [_ u  /  y ]_ [_ v  /  x ]_ C
9 csbeq1a 3439 . . . . 5  |-  ( y  =  u  ->  A  =  [_ u  /  y ]_ A )
10 csbeq1a 3439 . . . . . 6  |-  ( x  =  v  ->  C  =  [_ v  /  x ]_ C )
11 csbeq1a 3439 . . . . . 6  |-  ( y  =  u  ->  [_ v  /  x ]_ C  = 
[_ u  /  y ]_ [_ v  /  x ]_ C )
1210, 11sylan9eqr 2520 . . . . 5  |-  ( ( y  =  u  /\  x  =  v )  ->  C  =  [_ u  /  y ]_ [_ v  /  x ]_ C )
131, 2, 3, 4, 7, 8, 9, 12cbvmpt2x2 33069 . . . 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 3112 . . . . . . . 8  |-  v  e. 
_V
16 vex 3112 . . . . . . . 8  |-  u  e. 
_V
1715, 16op2ndd 6810 . . . . . . 7  |-  ( t  =  <. v ,  u >.  ->  ( 2nd `  t
)  =  u )
1817csbeq1d 3437 . . . . . 6  |-  ( t  =  <. v ,  u >.  ->  [_ ( 2nd `  t
)  /  y ]_ [_ ( 1st `  t
)  /  x ]_ C  =  [_ u  / 
y ]_ [_ ( 1st `  t )  /  x ]_ C )
1915, 16op1std 6809 . . . . . . . 8  |-  ( t  =  <. v ,  u >.  ->  ( 1st `  t
)  =  v )
2019csbeq1d 3437 . . . . . . 7  |-  ( t  =  <. v ,  u >.  ->  [_ ( 1st `  t
)  /  x ]_ C  =  [_ v  /  x ]_ C )
2120csbeq2dv 3843 . . . . . 6  |-  ( t  =  <. v ,  u >.  ->  [_ u  /  y ]_ [_ ( 1st `  t
)  /  x ]_ C  =  [_ u  / 
y ]_ [_ v  /  x ]_ C )
2218, 21eqtrd 2498 . . . . 5  |-  ( t  =  <. v ,  u >.  ->  [_ ( 2nd `  t
)  /  y ]_ [_ ( 1st `  t
)  /  x ]_ C  =  [_ u  / 
y ]_ [_ v  /  x ]_ C )
2322mpt2mptx2 33068 . . . 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 2496 . . 3  |-  F  =  ( t  e.  U_ u  e.  B  ( [_ u  /  y ]_ A  X.  { u } )  |->  [_ ( 2nd `  t )  / 
y ]_ [_ ( 1st `  t )  /  x ]_ C )
2524dmmptss 5509 . 2  |-  dom  F  C_ 
U_ u  e.  B  ( [_ u  /  y ]_ A  X.  { u } )
26 nfcv 2619 . . 3  |-  F/_ u
( A  X.  {
y } )
27 nfcv 2619 . . . 4  |-  F/_ y { u }
282, 27nfxp 5035 . . 3  |-  F/_ y
( [_ u  /  y ]_ A  X.  { u } )
29 sneq 4042 . . . 4  |-  ( y  =  u  ->  { y }  =  { u } )
309, 29xpeq12d 5033 . . 3  |-  ( y  =  u  ->  ( A  X.  { y } )  =  ( [_ u  /  y ]_ A  X.  { u } ) )
3126, 28, 30cbviun 4369 . 2  |-  U_ y  e.  B  ( A  X.  { y } )  =  U_ u  e.  B  ( [_ u  /  y ]_ A  X.  { u } )
3225, 31sseqtr4i 3532 1  |-  dom  F  C_ 
U_ y  e.  B  ( A  X.  { y } )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1395   [_csb 3430    C_ wss 3471   {csn 4032   <.cop 4038   U_ciun 4332    |-> cmpt 4515    X. cxp 5006   dom cdm 5008   ` cfv 5594    |-> cmpt2 6298   1stc1st 6797   2ndc2nd 6798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fv 5602  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800
This theorem is referenced by:  mpt2exxg2  33071
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