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Theorem ovmptss 6666
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 6649 . . . 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 2463 . . 3  |-  F  =  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
43fvmptss 5794 . 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 2987 . . . . . . . 8  |-  u  e. 
_V
6 vex 2987 . . . . . . . 8  |-  v  e. 
_V
75, 6op1std 6599 . . . . . . 7  |-  ( z  =  <. u ,  v
>.  ->  ( 1st `  z
)  =  u )
87csbeq1d 3307 . . . . . 6  |-  ( z  =  <. u ,  v
>.  ->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  =  [_ u  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
95, 6op2ndd 6600 . . . . . . . 8  |-  ( z  =  <. u ,  v
>.  ->  ( 2nd `  z
)  =  v )
109csbeq1d 3307 . . . . . . 7  |-  ( z  =  <. u ,  v
>.  ->  [_ ( 2nd `  z
)  /  y ]_ C  =  [_ v  / 
y ]_ C )
1110csbeq2dv 3699 . . . . . 6  |-  ( z  =  <. u ,  v
>.  ->  [_ u  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
128, 11eqtrd 2475 . . . . 5  |-  ( z  =  <. u ,  v
>.  ->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
1312sseq1d 3395 . . . 4  |-  ( z  =  <. u ,  v
>.  ->  ( [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C  C_  X  <->  [_ u  /  x ]_ [_ v  / 
y ]_ C  C_  X
) )
1413raliunxp 4991 . . 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 2589 . . . . 5  |-  F/_ u
( { x }  X.  B )
16 nfcv 2589 . . . . . 6  |-  F/_ x { u }
17 nfcsb1v 3316 . . . . . 6  |-  F/_ x [_ u  /  x ]_ B
1816, 17nfxp 4878 . . . . 5  |-  F/_ x
( { u }  X.  [_ u  /  x ]_ B )
19 sneq 3899 . . . . . 6  |-  ( x  =  u  ->  { x }  =  { u } )
20 csbeq1a 3309 . . . . . 6  |-  ( x  =  u  ->  B  =  [_ u  /  x ]_ B )
2119, 20xpeq12d 4877 . . . . 5  |-  ( x  =  u  ->  ( { x }  X.  B )  =  ( { u }  X.  [_ u  /  x ]_ B ) )
2215, 18, 21cbviun 4219 . . . 4  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  U_ u  e.  A  ( {
u }  X.  [_ u  /  x ]_ B
)
2322raleqi 2933 . . 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 1673 . . . 4  |-  F/ u A. y  e.  B  C  C_  X
25 nfcsb1v 3316 . . . . . 6  |-  F/_ x [_ u  /  x ]_ [_ v  /  y ]_ C
26 nfcv 2589 . . . . . 6  |-  F/_ x X
2725, 26nfss 3361 . . . . 5  |-  F/ x [_ u  /  x ]_ [_ v  /  y ]_ C  C_  X
2817, 27nfral 2781 . . . 4  |-  F/ x A. v  e.  [_  u  /  x ]_ B [_ u  /  x ]_ [_ v  /  y ]_ C  C_  X
29 nfv 1673 . . . . . 6  |-  F/ v  C  C_  X
30 nfcsb1v 3316 . . . . . . 7  |-  F/_ y [_ v  /  y ]_ C
31 nfcv 2589 . . . . . . 7  |-  F/_ y X
3230, 31nfss 3361 . . . . . 6  |-  F/ y
[_ v  /  y ]_ C  C_  X
33 csbeq1a 3309 . . . . . . 7  |-  ( y  =  v  ->  C  =  [_ v  /  y ]_ C )
3433sseq1d 3395 . . . . . 6  |-  ( y  =  v  ->  ( C  C_  X  <->  [_ v  / 
y ]_ C  C_  X
) )
3529, 32, 34cbvral 2955 . . . . 5  |-  ( A. y  e.  B  C  C_  X  <->  A. v  e.  B  [_ v  /  y ]_ C  C_  X )
36 csbeq1a 3309 . . . . . . 7  |-  ( x  =  u  ->  [_ v  /  y ]_ C  =  [_ u  /  x ]_ [_ v  /  y ]_ C )
3736sseq1d 3395 . . . . . 6  |-  ( x  =  u  ->  ( [_ v  /  y ]_ C  C_  X  <->  [_ u  /  x ]_ [_ v  / 
y ]_ C  C_  X
) )
3820, 37raleqbidv 2943 . . . . 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 2955 . . 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 6106 . . 3  |-  ( E F G )  =  ( F `  <. E ,  G >. )
4342sseq1i 3392 . 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 1369   A.wral 2727   [_csb 3300    C_ wss 3340   {csn 3889   <.cop 3895   U_ciun 4183    e. cmpt 4362    X. cxp 4850   ` cfv 5430  (class class class)co 6103    e. cmpt2 6105   1stc1st 6587   2ndc2nd 6588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-1st 6589  df-2nd 6590
This theorem is referenced by:  relmpt2opab  6667  relxpchom  15003  reldv  21357
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