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Theorem fvmptssdm 5350
Description: If all the values of the mapping are subsets of a class  C, then so is any evaluation of the mapping at a value in the domain of the mapping. (Contributed by Jim Kingdon, 3-Jan-2018.)
Hypothesis
Ref Expression
fvmpt2.1  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
fvmptssdm  |-  ( ( D  e.  dom  F  /\  A. x  e.  A  B  C_  C )  -> 
( F `  D
)  C_  C )
Distinct variable groups:    x, A    x, C
Allowed substitution hints:    B( x)    D( x)    F( x)

Proof of Theorem fvmptssdm
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fveq2 5268 . . . . . 6  |-  ( y  =  D  ->  ( F `  y )  =  ( F `  D ) )
21sseq1d 3042 . . . . 5  |-  ( y  =  D  ->  (
( F `  y
)  C_  C  <->  ( F `  D )  C_  C
) )
32imbi2d 228 . . . 4  |-  ( y  =  D  ->  (
( A. x  e.  A  B  C_  C  ->  ( F `  y
)  C_  C )  <->  ( A. x  e.  A  B  C_  C  ->  ( F `  D )  C_  C ) ) )
4 nfrab1 2542 . . . . . . 7  |-  F/_ x { x  e.  A  |  B  e.  _V }
54nfcri 2219 . . . . . 6  |-  F/ x  y  e.  { x  e.  A  |  B  e.  _V }
6 nfra1 2405 . . . . . . 7  |-  F/ x A. x  e.  A  B  C_  C
7 fvmpt2.1 . . . . . . . . . 10  |-  F  =  ( x  e.  A  |->  B )
8 nfmpt1 3906 . . . . . . . . . 10  |-  F/_ x
( x  e.  A  |->  B )
97, 8nfcxfr 2222 . . . . . . . . 9  |-  F/_ x F
10 nfcv 2225 . . . . . . . . 9  |-  F/_ x
y
119, 10nffv 5278 . . . . . . . 8  |-  F/_ x
( F `  y
)
12 nfcv 2225 . . . . . . . 8  |-  F/_ x C
1311, 12nfss 3007 . . . . . . 7  |-  F/ x
( F `  y
)  C_  C
146, 13nfim 1507 . . . . . 6  |-  F/ x
( A. x  e.  A  B  C_  C  ->  ( F `  y
)  C_  C )
155, 14nfim 1507 . . . . 5  |-  F/ x
( y  e.  {
x  e.  A  |  B  e.  _V }  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  y )  C_  C ) )
16 eleq1 2147 . . . . . 6  |-  ( x  =  y  ->  (
x  e.  { x  e.  A  |  B  e.  _V }  <->  y  e.  { x  e.  A  |  B  e.  _V } ) )
17 fveq2 5268 . . . . . . . 8  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
1817sseq1d 3042 . . . . . . 7  |-  ( x  =  y  ->  (
( F `  x
)  C_  C  <->  ( F `  y )  C_  C
) )
1918imbi2d 228 . . . . . 6  |-  ( x  =  y  ->  (
( A. x  e.  A  B  C_  C  ->  ( F `  x
)  C_  C )  <->  ( A. x  e.  A  B  C_  C  ->  ( F `  y )  C_  C ) ) )
2016, 19imbi12d 232 . . . . 5  |-  ( x  =  y  ->  (
( x  e.  {
x  e.  A  |  B  e.  _V }  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  x )  C_  C ) )  <->  ( y  e.  { x  e.  A  |  B  e.  _V }  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  y
)  C_  C )
) ) )
217dmmpt 4892 . . . . . . 7  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
2221eleq2i 2151 . . . . . 6  |-  ( x  e.  dom  F  <->  x  e.  { x  e.  A  |  B  e.  _V } )
2321rabeq2i 2612 . . . . . . . . . 10  |-  ( x  e.  dom  F  <->  ( x  e.  A  /\  B  e. 
_V ) )
247fvmpt2 5349 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  B  e.  _V )  ->  ( F `  x
)  =  B )
25 eqimss 3067 . . . . . . . . . . 11  |-  ( ( F `  x )  =  B  ->  ( F `  x )  C_  B )
2624, 25syl 14 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  B  e.  _V )  ->  ( F `  x
)  C_  B )
2723, 26sylbi 119 . . . . . . . . 9  |-  ( x  e.  dom  F  -> 
( F `  x
)  C_  B )
2827adantr 270 . . . . . . . 8  |-  ( ( x  e.  dom  F  /\  A. x  e.  A  B  C_  C )  -> 
( F `  x
)  C_  B )
297dmmptss 4893 . . . . . . . . . 10  |-  dom  F  C_  A
3029sseli 3010 . . . . . . . . 9  |-  ( x  e.  dom  F  ->  x  e.  A )
31 rsp 2419 . . . . . . . . 9  |-  ( A. x  e.  A  B  C_  C  ->  ( x  e.  A  ->  B  C_  C ) )
3230, 31mpan9 275 . . . . . . . 8  |-  ( ( x  e.  dom  F  /\  A. x  e.  A  B  C_  C )  ->  B  C_  C )
3328, 32sstrd 3024 . . . . . . 7  |-  ( ( x  e.  dom  F  /\  A. x  e.  A  B  C_  C )  -> 
( F `  x
)  C_  C )
3433ex 113 . . . . . 6  |-  ( x  e.  dom  F  -> 
( A. x  e.  A  B  C_  C  ->  ( F `  x
)  C_  C )
)
3522, 34sylbir 133 . . . . 5  |-  ( x  e.  { x  e.  A  |  B  e. 
_V }  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  x )  C_  C ) )
3615, 20, 35chvar 1684 . . . 4  |-  ( y  e.  { x  e.  A  |  B  e. 
_V }  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  y )  C_  C ) )
373, 36vtoclga 2678 . . 3  |-  ( D  e.  { x  e.  A  |  B  e. 
_V }  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  D )  C_  C ) )
3837, 21eleq2s 2179 . 2  |-  ( D  e.  dom  F  -> 
( A. x  e.  A  B  C_  C  ->  ( F `  D
)  C_  C )
)
3938imp 122 1  |-  ( ( D  e.  dom  F  /\  A. x  e.  A  B  C_  C )  -> 
( F `  D
)  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1287    e. wcel 1436   A.wral 2355   {crab 2359   _Vcvv 2615    C_ wss 2988    |-> cmpt 3874   dom cdm 4411   ` cfv 4981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-mpt 3876  df-id 4094  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fv 4989
This theorem is referenced by: (None)
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