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Theorem fvmptssdm 5553
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 5469 . . . . . 6  |-  ( y  =  D  ->  ( F `  y )  =  ( F `  D ) )
21sseq1d 3157 . . . . 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 2636 . . . . . . 7  |-  F/_ x { x  e.  A  |  B  e.  _V }
54nfcri 2293 . . . . . 6  |-  F/ x  y  e.  { x  e.  A  |  B  e.  _V }
6 nfra1 2488 . . . . . . 7  |-  F/ x A. x  e.  A  B  C_  C
7 fvmpt2.1 . . . . . . . . . 10  |-  F  =  ( x  e.  A  |->  B )
8 nfmpt1 4058 . . . . . . . . . 10  |-  F/_ x
( x  e.  A  |->  B )
97, 8nfcxfr 2296 . . . . . . . . 9  |-  F/_ x F
10 nfcv 2299 . . . . . . . . 9  |-  F/_ x
y
119, 10nffv 5479 . . . . . . . 8  |-  F/_ x
( F `  y
)
12 nfcv 2299 . . . . . . . 8  |-  F/_ x C
1311, 12nfss 3121 . . . . . . 7  |-  F/ x
( F `  y
)  C_  C
146, 13nfim 1552 . . . . . 6  |-  F/ x
( A. x  e.  A  B  C_  C  ->  ( F `  y
)  C_  C )
155, 14nfim 1552 . . . . 5  |-  F/ x
( y  e.  {
x  e.  A  |  B  e.  _V }  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  y )  C_  C ) )
16 eleq1 2220 . . . . . 6  |-  ( x  =  y  ->  (
x  e.  { x  e.  A  |  B  e.  _V }  <->  y  e.  { x  e.  A  |  B  e.  _V } ) )
17 fveq2 5469 . . . . . . . 8  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
1817sseq1d 3157 . . . . . . 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 5082 . . . . . . 7  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
2221eleq2i 2224 . . . . . 6  |-  ( x  e.  dom  F  <->  x  e.  { x  e.  A  |  B  e.  _V } )
2321rabeq2i 2709 . . . . . . . . . 10  |-  ( x  e.  dom  F  <->  ( x  e.  A  /\  B  e. 
_V ) )
247fvmpt2 5552 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  B  e.  _V )  ->  ( F `  x
)  =  B )
25 eqimss 3182 . . . . . . . . . . 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 5083 . . . . . . . . . 10  |-  dom  F  C_  A
3029sseli 3124 . . . . . . . . 9  |-  ( x  e.  dom  F  ->  x  e.  A )
31 rsp 2504 . . . . . . . . 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 3138 . . . . . . 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 1737 . . . 4  |-  ( y  e.  { x  e.  A  |  B  e. 
_V }  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  y )  C_  C ) )
373, 36vtoclga 2778 . . 3  |-  ( D  e.  { x  e.  A  |  B  e. 
_V }  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  D )  C_  C ) )
3837, 21eleq2s 2252 . 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 1335    e. wcel 2128   A.wral 2435   {crab 2439   _Vcvv 2712    C_ wss 3102    |-> cmpt 4026   dom cdm 4587   ` cfv 5171
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-mpt 4028  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fv 5179
This theorem is referenced by: (None)
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