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Theorem ovmptss 6200
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 6187 . . . 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 2303 . . 3  |-  F  =  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
43fvmptss 5609 . 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 2791 . . . . . . . 8  |-  u  e. 
_V
6 vex 2791 . . . . . . . 8  |-  v  e. 
_V
75, 6op1std 6130 . . . . . . 7  |-  ( z  =  <. u ,  v
>.  ->  ( 1st `  z
)  =  u )
87csbeq1d 3087 . . . . . 6  |-  ( z  =  <. u ,  v
>.  ->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  =  [_ u  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
95, 6op2ndd 6131 . . . . . . . 8  |-  ( z  =  <. u ,  v
>.  ->  ( 2nd `  z
)  =  v )
109csbeq1d 3087 . . . . . . 7  |-  ( z  =  <. u ,  v
>.  ->  [_ ( 2nd `  z
)  /  y ]_ C  =  [_ v  / 
y ]_ C )
1110csbeq2dv 3106 . . . . . 6  |-  ( z  =  <. u ,  v
>.  ->  [_ u  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
128, 11eqtrd 2315 . . . . 5  |-  ( z  =  <. u ,  v
>.  ->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
1312sseq1d 3205 . . . 4  |-  ( z  =  <. u ,  v
>.  ->  ( [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C  C_  X  <->  [_ u  /  x ]_ [_ v  / 
y ]_ C  C_  X
) )
1413raliunxp 4825 . . 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 2419 . . . . 5  |-  F/_ u
( { x }  X.  B )
16 nfcv 2419 . . . . . 6  |-  F/_ x { u }
17 nfcsb1v 3113 . . . . . 6  |-  F/_ x [_ u  /  x ]_ B
1816, 17nfxp 4715 . . . . 5  |-  F/_ x
( { u }  X.  [_ u  /  x ]_ B )
19 sneq 3651 . . . . . 6  |-  ( x  =  u  ->  { x }  =  { u } )
20 csbeq1a 3089 . . . . . 6  |-  ( x  =  u  ->  B  =  [_ u  /  x ]_ B )
2119, 20xpeq12d 4714 . . . . 5  |-  ( x  =  u  ->  ( { x }  X.  B )  =  ( { u }  X.  [_ u  /  x ]_ B ) )
2215, 18, 21cbviun 3939 . . . 4  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  U_ u  e.  A  ( {
u }  X.  [_ u  /  x ]_ B
)
2322raleqi 2740 . . 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 1605 . . . 4  |-  F/ u A. y  e.  B  C  C_  X
25 nfcsb1v 3113 . . . . . 6  |-  F/_ x [_ u  /  x ]_ [_ v  /  y ]_ C
26 nfcv 2419 . . . . . 6  |-  F/_ x X
2725, 26nfss 3173 . . . . 5  |-  F/ x [_ u  /  x ]_ [_ v  /  y ]_ C  C_  X
2817, 27nfral 2596 . . . 4  |-  F/ x A. v  e.  [_  u  /  x ]_ B [_ u  /  x ]_ [_ v  /  y ]_ C  C_  X
29 nfv 1605 . . . . . 6  |-  F/ v  C  C_  X
30 nfcsb1v 3113 . . . . . . 7  |-  F/_ y [_ v  /  y ]_ C
31 nfcv 2419 . . . . . . 7  |-  F/_ y X
3230, 31nfss 3173 . . . . . 6  |-  F/ y
[_ v  /  y ]_ C  C_  X
33 csbeq1a 3089 . . . . . . 7  |-  ( y  =  v  ->  C  =  [_ v  /  y ]_ C )
3433sseq1d 3205 . . . . . 6  |-  ( y  =  v  ->  ( C  C_  X  <->  [_ v  / 
y ]_ C  C_  X
) )
3529, 32, 34cbvral 2760 . . . . 5  |-  ( A. y  e.  B  C  C_  X  <->  A. v  e.  B  [_ v  /  y ]_ C  C_  X )
36 csbeq1a 3089 . . . . . . 7  |-  ( x  =  u  ->  [_ v  /  y ]_ C  =  [_ u  /  x ]_ [_ v  /  y ]_ C )
3736sseq1d 3205 . . . . . 6  |-  ( x  =  u  ->  ( [_ v  /  y ]_ C  C_  X  <->  [_ u  /  x ]_ [_ v  / 
y ]_ C  C_  X
) )
3820, 37raleqbidv 2748 . . . . 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 248 . . . 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 2760 . . 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 269 . 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 5861 . . 3  |-  ( E F G )  =  ( F `  <. E ,  G >. )
4342sseq1i 3202 . 2  |-  ( ( E F G ) 
C_  X  <->  ( F `  <. E ,  G >. )  C_  X )
444, 41, 433imtr4i 257 1  |-  ( A. x  e.  A  A. y  e.  B  C  C_  X  ->  ( E F G )  C_  X
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   A.wral 2543   [_csb 3081    C_ wss 3152   {csn 3640   <.cop 3643   U_ciun 3905    e. cmpt 4077    X. cxp 4687   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121
This theorem is referenced by:  relmpt2opab  6201  relxpchom  13955  reldv  19220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123
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