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Theorem fvmptssdm 5601
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 5516 . . . . . 6  |-  ( y  =  D  ->  ( F `  y )  =  ( F `  D ) )
21sseq1d 3185 . . . . 5  |-  ( y  =  D  ->  (
( F `  y
)  C_  C  <->  ( F `  D )  C_  C
) )
32imbi2d 230 . . . 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 2657 . . . . . . 7  |-  F/_ x { x  e.  A  |  B  e.  _V }
54nfcri 2313 . . . . . 6  |-  F/ x  y  e.  { x  e.  A  |  B  e.  _V }
6 nfra1 2508 . . . . . . 7  |-  F/ x A. x  e.  A  B  C_  C
7 fvmpt2.1 . . . . . . . . . 10  |-  F  =  ( x  e.  A  |->  B )
8 nfmpt1 4097 . . . . . . . . . 10  |-  F/_ x
( x  e.  A  |->  B )
97, 8nfcxfr 2316 . . . . . . . . 9  |-  F/_ x F
10 nfcv 2319 . . . . . . . . 9  |-  F/_ x
y
119, 10nffv 5526 . . . . . . . 8  |-  F/_ x
( F `  y
)
12 nfcv 2319 . . . . . . . 8  |-  F/_ x C
1311, 12nfss 3149 . . . . . . 7  |-  F/ x
( F `  y
)  C_  C
146, 13nfim 1572 . . . . . 6  |-  F/ x
( A. x  e.  A  B  C_  C  ->  ( F `  y
)  C_  C )
155, 14nfim 1572 . . . . 5  |-  F/ x
( y  e.  {
x  e.  A  |  B  e.  _V }  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  y )  C_  C ) )
16 eleq1 2240 . . . . . 6  |-  ( x  =  y  ->  (
x  e.  { x  e.  A  |  B  e.  _V }  <->  y  e.  { x  e.  A  |  B  e.  _V } ) )
17 fveq2 5516 . . . . . . . 8  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
1817sseq1d 3185 . . . . . . 7  |-  ( x  =  y  ->  (
( F `  x
)  C_  C  <->  ( F `  y )  C_  C
) )
1918imbi2d 230 . . . . . 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 234 . . . . 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 5125 . . . . . . 7  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
2221eleq2i 2244 . . . . . 6  |-  ( x  e.  dom  F  <->  x  e.  { x  e.  A  |  B  e.  _V } )
2321rabeq2i 2735 . . . . . . . . . 10  |-  ( x  e.  dom  F  <->  ( x  e.  A  /\  B  e. 
_V ) )
247fvmpt2 5600 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  B  e.  _V )  ->  ( F `  x
)  =  B )
25 eqimss 3210 . . . . . . . . . . 11  |-  ( ( F `  x )  =  B  ->  ( F `  x )  C_  B )
2624, 25syl 14 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  B  e.  _V )  ->  ( F `  x
)  C_  B )
2723, 26sylbi 121 . . . . . . . . 9  |-  ( x  e.  dom  F  -> 
( F `  x
)  C_  B )
2827adantr 276 . . . . . . . 8  |-  ( ( x  e.  dom  F  /\  A. x  e.  A  B  C_  C )  -> 
( F `  x
)  C_  B )
297dmmptss 5126 . . . . . . . . . 10  |-  dom  F  C_  A
3029sseli 3152 . . . . . . . . 9  |-  ( x  e.  dom  F  ->  x  e.  A )
31 rsp 2524 . . . . . . . . 9  |-  ( A. x  e.  A  B  C_  C  ->  ( x  e.  A  ->  B  C_  C ) )
3230, 31mpan9 281 . . . . . . . 8  |-  ( ( x  e.  dom  F  /\  A. x  e.  A  B  C_  C )  ->  B  C_  C )
3328, 32sstrd 3166 . . . . . . 7  |-  ( ( x  e.  dom  F  /\  A. x  e.  A  B  C_  C )  -> 
( F `  x
)  C_  C )
3433ex 115 . . . . . 6  |-  ( x  e.  dom  F  -> 
( A. x  e.  A  B  C_  C  ->  ( F `  x
)  C_  C )
)
3522, 34sylbir 135 . . . . 5  |-  ( x  e.  { x  e.  A  |  B  e. 
_V }  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  x )  C_  C ) )
3615, 20, 35chvar 1757 . . . 4  |-  ( y  e.  { x  e.  A  |  B  e. 
_V }  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  y )  C_  C ) )
373, 36vtoclga 2804 . . 3  |-  ( D  e.  { x  e.  A  |  B  e. 
_V }  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  D )  C_  C ) )
3837, 21eleq2s 2272 . 2  |-  ( D  e.  dom  F  -> 
( A. x  e.  A  B  C_  C  ->  ( F `  D
)  C_  C )
)
3938imp 124 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 104    = wceq 1353    e. wcel 2148   A.wral 2455   {crab 2459   _Vcvv 2738    C_ wss 3130    |-> cmpt 4065   dom cdm 4627   ` cfv 5217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fv 5225
This theorem is referenced by: (None)
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