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Theorem fvmptssdm 5283
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 5206 . . . . . 6  |-  ( y  =  D  ->  ( F `  y )  =  ( F `  D ) )
21sseq1d 3000 . . . . 5  |-  ( y  =  D  ->  (
( F `  y
)  C_  C  <->  ( F `  D )  C_  C
) )
32imbi2d 223 . . . 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 2506 . . . . . . 7  |-  F/_ x { x  e.  A  |  B  e.  _V }
54nfcri 2188 . . . . . 6  |-  F/ x  y  e.  { x  e.  A  |  B  e.  _V }
6 nfra1 2372 . . . . . . 7  |-  F/ x A. x  e.  A  B  C_  C
7 fvmpt2.1 . . . . . . . . . 10  |-  F  =  ( x  e.  A  |->  B )
8 nfmpt1 3878 . . . . . . . . . 10  |-  F/_ x
( x  e.  A  |->  B )
97, 8nfcxfr 2191 . . . . . . . . 9  |-  F/_ x F
10 nfcv 2194 . . . . . . . . 9  |-  F/_ x
y
119, 10nffv 5213 . . . . . . . 8  |-  F/_ x
( F `  y
)
12 nfcv 2194 . . . . . . . 8  |-  F/_ x C
1311, 12nfss 2966 . . . . . . 7  |-  F/ x
( F `  y
)  C_  C
146, 13nfim 1480 . . . . . 6  |-  F/ x
( A. x  e.  A  B  C_  C  ->  ( F `  y
)  C_  C )
155, 14nfim 1480 . . . . 5  |-  F/ x
( y  e.  {
x  e.  A  |  B  e.  _V }  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  y )  C_  C ) )
16 eleq1 2116 . . . . . 6  |-  ( x  =  y  ->  (
x  e.  { x  e.  A  |  B  e.  _V }  <->  y  e.  { x  e.  A  |  B  e.  _V } ) )
17 fveq2 5206 . . . . . . . 8  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
1817sseq1d 3000 . . . . . . 7  |-  ( x  =  y  ->  (
( F `  x
)  C_  C  <->  ( F `  y )  C_  C
) )
1918imbi2d 223 . . . . . 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 227 . . . . 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 4844 . . . . . . 7  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
2221eleq2i 2120 . . . . . 6  |-  ( x  e.  dom  F  <->  x  e.  { x  e.  A  |  B  e.  _V } )
2321rabeq2i 2571 . . . . . . . . . 10  |-  ( x  e.  dom  F  <->  ( x  e.  A  /\  B  e. 
_V ) )
247fvmpt2 5282 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  B  e.  _V )  ->  ( F `  x
)  =  B )
25 eqimss 3025 . . . . . . . . . . 11  |-  ( ( F `  x )  =  B  ->  ( F `  x )  C_  B )
2624, 25syl 14 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  B  e.  _V )  ->  ( F `  x
)  C_  B )
2723, 26sylbi 118 . . . . . . . . 9  |-  ( x  e.  dom  F  -> 
( F `  x
)  C_  B )
2827adantr 265 . . . . . . . 8  |-  ( ( x  e.  dom  F  /\  A. x  e.  A  B  C_  C )  -> 
( F `  x
)  C_  B )
297dmmptss 4845 . . . . . . . . . 10  |-  dom  F  C_  A
3029sseli 2969 . . . . . . . . 9  |-  ( x  e.  dom  F  ->  x  e.  A )
31 rsp 2386 . . . . . . . . 9  |-  ( A. x  e.  A  B  C_  C  ->  ( x  e.  A  ->  B  C_  C ) )
3230, 31mpan9 269 . . . . . . . 8  |-  ( ( x  e.  dom  F  /\  A. x  e.  A  B  C_  C )  ->  B  C_  C )
3328, 32sstrd 2983 . . . . . . 7  |-  ( ( x  e.  dom  F  /\  A. x  e.  A  B  C_  C )  -> 
( F `  x
)  C_  C )
3433ex 112 . . . . . 6  |-  ( x  e.  dom  F  -> 
( A. x  e.  A  B  C_  C  ->  ( F `  x
)  C_  C )
)
3522, 34sylbir 129 . . . . 5  |-  ( x  e.  { x  e.  A  |  B  e. 
_V }  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  x )  C_  C ) )
3615, 20, 35chvar 1656 . . . 4  |-  ( y  e.  { x  e.  A  |  B  e. 
_V }  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  y )  C_  C ) )
373, 36vtoclga 2636 . . 3  |-  ( D  e.  { x  e.  A  |  B  e. 
_V }  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  D )  C_  C ) )
3837, 21eleq2s 2148 . 2  |-  ( D  e.  dom  F  -> 
( A. x  e.  A  B  C_  C  ->  ( F `  D
)  C_  C )
)
3938imp 119 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 101    = wceq 1259    e. wcel 1409   A.wral 2323   {crab 2327   _Vcvv 2574    C_ wss 2945    |-> cmpt 3846   dom cdm 4373   ` cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fv 4938
This theorem is referenced by: (None)
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