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Theorem ovmptss 6854
Description: If all the values of the mapping are subsets of a class  X, then so is any evaluation of the mapping. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
ovmptss.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
ovmptss  |-  ( A. x  e.  A  A. y  e.  B  C  C_  X  ->  ( E F G )  C_  X
)
Distinct variable groups:    x, y, A    y, B    x, X, y
Allowed substitution hints:    B( x)    C( x, y)    E( x, y)    F( x, y)    G( x, y)

Proof of Theorem ovmptss
Dummy variables  v  u  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovmptss.1 . . . 4  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
2 mpt2mptsx 6836 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e. 
U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
31, 2eqtri 2483 . . 3  |-  F  =  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
43fvmptss 5940 . 2  |-  ( A. z  e.  U_  x  e.  A  ( { x }  X.  B ) [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  / 
y ]_ C  C_  X  ->  ( F `  <. E ,  G >. )  C_  X )
5 vex 3109 . . . . . . . 8  |-  u  e. 
_V
6 vex 3109 . . . . . . . 8  |-  v  e. 
_V
75, 6op1std 6783 . . . . . . 7  |-  ( z  =  <. u ,  v
>.  ->  ( 1st `  z
)  =  u )
87csbeq1d 3427 . . . . . 6  |-  ( z  =  <. u ,  v
>.  ->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  =  [_ u  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
95, 6op2ndd 6784 . . . . . . . 8  |-  ( z  =  <. u ,  v
>.  ->  ( 2nd `  z
)  =  v )
109csbeq1d 3427 . . . . . . 7  |-  ( z  =  <. u ,  v
>.  ->  [_ ( 2nd `  z
)  /  y ]_ C  =  [_ v  / 
y ]_ C )
1110csbeq2dv 3831 . . . . . 6  |-  ( z  =  <. u ,  v
>.  ->  [_ u  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
128, 11eqtrd 2495 . . . . 5  |-  ( z  =  <. u ,  v
>.  ->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
1312sseq1d 3516 . . . 4  |-  ( z  =  <. u ,  v
>.  ->  ( [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C  C_  X  <->  [_ u  /  x ]_ [_ v  / 
y ]_ C  C_  X
) )
1413raliunxp 5131 . . 3  |-  ( A. z  e.  U_  u  e.  A  ( { u }  X.  [_ u  /  x ]_ B ) [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  / 
y ]_ C  C_  X  <->  A. u  e.  A  A. v  e.  [_  u  /  x ]_ B [_ u  /  x ]_ [_ v  /  y ]_ C  C_  X )
15 nfcv 2616 . . . . 5  |-  F/_ u
( { x }  X.  B )
16 nfcv 2616 . . . . . 6  |-  F/_ x { u }
17 nfcsb1v 3436 . . . . . 6  |-  F/_ x [_ u  /  x ]_ B
1816, 17nfxp 5015 . . . . 5  |-  F/_ x
( { u }  X.  [_ u  /  x ]_ B )
19 sneq 4026 . . . . . 6  |-  ( x  =  u  ->  { x }  =  { u } )
20 csbeq1a 3429 . . . . . 6  |-  ( x  =  u  ->  B  =  [_ u  /  x ]_ B )
2119, 20xpeq12d 5013 . . . . 5  |-  ( x  =  u  ->  ( { x }  X.  B )  =  ( { u }  X.  [_ u  /  x ]_ B ) )
2215, 18, 21cbviun 4352 . . . 4  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  U_ u  e.  A  ( {
u }  X.  [_ u  /  x ]_ B
)
2322raleqi 3055 . . 3  |-  ( A. z  e.  U_  x  e.  A  ( { x }  X.  B ) [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  / 
y ]_ C  C_  X  <->  A. z  e.  U_  u  e.  A  ( {
u }  X.  [_ u  /  x ]_ B
) [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  C_  X )
24 nfv 1712 . . . 4  |-  F/ u A. y  e.  B  C  C_  X
25 nfcsb1v 3436 . . . . . 6  |-  F/_ x [_ u  /  x ]_ [_ v  /  y ]_ C
26 nfcv 2616 . . . . . 6  |-  F/_ x X
2725, 26nfss 3482 . . . . 5  |-  F/ x [_ u  /  x ]_ [_ v  /  y ]_ C  C_  X
2817, 27nfral 2840 . . . 4  |-  F/ x A. v  e.  [_  u  /  x ]_ B [_ u  /  x ]_ [_ v  /  y ]_ C  C_  X
29 nfv 1712 . . . . . 6  |-  F/ v  C  C_  X
30 nfcsb1v 3436 . . . . . . 7  |-  F/_ y [_ v  /  y ]_ C
31 nfcv 2616 . . . . . . 7  |-  F/_ y X
3230, 31nfss 3482 . . . . . 6  |-  F/ y
[_ v  /  y ]_ C  C_  X
33 csbeq1a 3429 . . . . . . 7  |-  ( y  =  v  ->  C  =  [_ v  /  y ]_ C )
3433sseq1d 3516 . . . . . 6  |-  ( y  =  v  ->  ( C  C_  X  <->  [_ v  / 
y ]_ C  C_  X
) )
3529, 32, 34cbvral 3077 . . . . 5  |-  ( A. y  e.  B  C  C_  X  <->  A. v  e.  B  [_ v  /  y ]_ C  C_  X )
36 csbeq1a 3429 . . . . . . 7  |-  ( x  =  u  ->  [_ v  /  y ]_ C  =  [_ u  /  x ]_ [_ v  /  y ]_ C )
3736sseq1d 3516 . . . . . 6  |-  ( x  =  u  ->  ( [_ v  /  y ]_ C  C_  X  <->  [_ u  /  x ]_ [_ v  / 
y ]_ C  C_  X
) )
3820, 37raleqbidv 3065 . . . . 5  |-  ( x  =  u  ->  ( A. v  e.  B  [_ v  /  y ]_ C  C_  X  <->  A. v  e.  [_  u  /  x ]_ B [_ u  /  x ]_ [_ v  / 
y ]_ C  C_  X
) )
3935, 38syl5bb 257 . . . 4  |-  ( x  =  u  ->  ( A. y  e.  B  C  C_  X  <->  A. v  e.  [_  u  /  x ]_ B [_ u  /  x ]_ [_ v  / 
y ]_ C  C_  X
) )
4024, 28, 39cbvral 3077 . . 3  |-  ( A. x  e.  A  A. y  e.  B  C  C_  X  <->  A. u  e.  A  A. v  e.  [_  u  /  x ]_ B [_ u  /  x ]_ [_ v  /  y ]_ C  C_  X )
4114, 23, 403bitr4ri 278 . 2  |-  ( A. x  e.  A  A. y  e.  B  C  C_  X  <->  A. z  e.  U_  x  e.  A  ( { x }  X.  B ) [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C  C_  X )
42 df-ov 6273 . . 3  |-  ( E F G )  =  ( F `  <. E ,  G >. )
4342sseq1i 3513 . 2  |-  ( ( E F G ) 
C_  X  <->  ( F `  <. E ,  G >. )  C_  X )
444, 41, 433imtr4i 266 1  |-  ( A. x  e.  A  A. y  e.  B  C  C_  X  ->  ( E F G )  C_  X
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398   A.wral 2804   [_csb 3420    C_ wss 3461   {csn 4016   <.cop 4022   U_ciun 4315    |-> cmpt 4497    X. cxp 4986   ` cfv 5570  (class class class)co 6270    |-> 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-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774
This theorem is referenced by:  relmpt2opab  6855  relxpchom  15652  reldv  22443
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