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Theorem fvmptssdm 5580
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 5496 . . . . . 6  |-  ( y  =  D  ->  ( F `  y )  =  ( F `  D ) )
21sseq1d 3176 . . . . 5  |-  ( y  =  D  ->  (
( F `  y
)  C_  C  <->  ( F `  D )  C_  C
) )
32imbi2d 229 . . . 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 2649 . . . . . . 7  |-  F/_ x { x  e.  A  |  B  e.  _V }
54nfcri 2306 . . . . . 6  |-  F/ x  y  e.  { x  e.  A  |  B  e.  _V }
6 nfra1 2501 . . . . . . 7  |-  F/ x A. x  e.  A  B  C_  C
7 fvmpt2.1 . . . . . . . . . 10  |-  F  =  ( x  e.  A  |->  B )
8 nfmpt1 4082 . . . . . . . . . 10  |-  F/_ x
( x  e.  A  |->  B )
97, 8nfcxfr 2309 . . . . . . . . 9  |-  F/_ x F
10 nfcv 2312 . . . . . . . . 9  |-  F/_ x
y
119, 10nffv 5506 . . . . . . . 8  |-  F/_ x
( F `  y
)
12 nfcv 2312 . . . . . . . 8  |-  F/_ x C
1311, 12nfss 3140 . . . . . . 7  |-  F/ x
( F `  y
)  C_  C
146, 13nfim 1565 . . . . . 6  |-  F/ x
( A. x  e.  A  B  C_  C  ->  ( F `  y
)  C_  C )
155, 14nfim 1565 . . . . 5  |-  F/ x
( y  e.  {
x  e.  A  |  B  e.  _V }  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  y )  C_  C ) )
16 eleq1 2233 . . . . . 6  |-  ( x  =  y  ->  (
x  e.  { x  e.  A  |  B  e.  _V }  <->  y  e.  { x  e.  A  |  B  e.  _V } ) )
17 fveq2 5496 . . . . . . . 8  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
1817sseq1d 3176 . . . . . . 7  |-  ( x  =  y  ->  (
( F `  x
)  C_  C  <->  ( F `  y )  C_  C
) )
1918imbi2d 229 . . . . . 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 233 . . . . 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 5106 . . . . . . 7  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
2221eleq2i 2237 . . . . . 6  |-  ( x  e.  dom  F  <->  x  e.  { x  e.  A  |  B  e.  _V } )
2321rabeq2i 2727 . . . . . . . . . 10  |-  ( x  e.  dom  F  <->  ( x  e.  A  /\  B  e. 
_V ) )
247fvmpt2 5579 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  B  e.  _V )  ->  ( F `  x
)  =  B )
25 eqimss 3201 . . . . . . . . . . 11  |-  ( ( F `  x )  =  B  ->  ( F `  x )  C_  B )
2624, 25syl 14 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  B  e.  _V )  ->  ( F `  x
)  C_  B )
2723, 26sylbi 120 . . . . . . . . 9  |-  ( x  e.  dom  F  -> 
( F `  x
)  C_  B )
2827adantr 274 . . . . . . . 8  |-  ( ( x  e.  dom  F  /\  A. x  e.  A  B  C_  C )  -> 
( F `  x
)  C_  B )
297dmmptss 5107 . . . . . . . . . 10  |-  dom  F  C_  A
3029sseli 3143 . . . . . . . . 9  |-  ( x  e.  dom  F  ->  x  e.  A )
31 rsp 2517 . . . . . . . . 9  |-  ( A. x  e.  A  B  C_  C  ->  ( x  e.  A  ->  B  C_  C ) )
3230, 31mpan9 279 . . . . . . . 8  |-  ( ( x  e.  dom  F  /\  A. x  e.  A  B  C_  C )  ->  B  C_  C )
3328, 32sstrd 3157 . . . . . . 7  |-  ( ( x  e.  dom  F  /\  A. x  e.  A  B  C_  C )  -> 
( F `  x
)  C_  C )
3433ex 114 . . . . . 6  |-  ( x  e.  dom  F  -> 
( A. x  e.  A  B  C_  C  ->  ( F `  x
)  C_  C )
)
3522, 34sylbir 134 . . . . 5  |-  ( x  e.  { x  e.  A  |  B  e. 
_V }  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  x )  C_  C ) )
3615, 20, 35chvar 1750 . . . 4  |-  ( y  e.  { x  e.  A  |  B  e. 
_V }  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  y )  C_  C ) )
373, 36vtoclga 2796 . . 3  |-  ( D  e.  { x  e.  A  |  B  e. 
_V }  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  D )  C_  C ) )
3837, 21eleq2s 2265 . 2  |-  ( D  e.  dom  F  -> 
( A. x  e.  A  B  C_  C  ->  ( F `  D
)  C_  C )
)
3938imp 123 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 103    = wceq 1348    e. wcel 2141   A.wral 2448   {crab 2452   _Vcvv 2730    C_ wss 3121    |-> cmpt 4050   dom cdm 4611   ` cfv 5198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fv 5206
This theorem is referenced by: (None)
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