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Theorem fvmptssdm 5198
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  |->
Assertion
Ref Expression
fvmptssdm  D  dom  F  C_  C  F `  D 
C_  C
Distinct variable groups:   ,   , C
Allowed substitution hints:   ()    D()    F()

Proof of Theorem fvmptssdm
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fveq2 5121 . . . . . 6  D  F `  F `  D
21sseq1d 2966 . . . . 5  D  F ` 
C_  C  F `
 D  C_  C
32imbi2d 219 . . . 4  D  C_  C  F `  C_  C  C_  C  F `
 D  C_  C
4 nfrab1 2483 . . . . . . 7  F/_ {  |  _V }
54nfcri 2169 . . . . . 6  F/  {  |  _V }
6 nfra1 2349 . . . . . . 7  F/  C_  C
7 fvmpt2.1 . . . . . . . . . 10  F  |->
8 nfmpt1 3841 . . . . . . . . . 10  F/_  |->
97, 8nfcxfr 2172 . . . . . . . . 9  F/_ F
10 nfcv 2175 . . . . . . . . 9  F/_
119, 10nffv 5128 . . . . . . . 8  F/_ F `
12 nfcv 2175 . . . . . . . 8  F/_ C
1311, 12nfss 2932 . . . . . . 7  F/ F ` 
C_  C
146, 13nfim 1461 . . . . . 6  F/  C_  C  F `  C_  C
155, 14nfim 1461 . . . . 5  F/  {  |  _V }  C_  C  F `
 C_  C
16 eleq1 2097 . . . . . 6  {  |  _V }  {  |  _V }
17 fveq2 5121 . . . . . . . 8  F `  F `
1817sseq1d 2966 . . . . . . 7  F ` 
C_  C  F `
 C_  C
1918imbi2d 219 . . . . . 6  C_  C  F `  C_  C  C_  C  F `
 C_  C
2016, 19imbi12d 223 . . . . 5  {  |  _V }  C_  C  F `
 C_  C  {  |  _V }  C_  C  F `
 C_  C
217dmmpt 4759 . . . . . . 7  dom  F  {  |  _V }
2221eleq2i 2101 . . . . . 6  dom  F 
{  |  _V }
2321rabeq2i 2548 . . . . . . . . . 10  dom  F  _V
247fvmpt2 5197 . . . . . . . . . . 11  _V  F `
25 eqimss 2991 . . . . . . . . . . 11  F `  F `  C_
2624, 25syl 14 . . . . . . . . . 10  _V  F `  C_
2723, 26sylbi 114 . . . . . . . . 9  dom  F  F ` 
C_
2827adantr 261 . . . . . . . 8  dom  F  C_  C  F ` 
C_
297dmmptss 4760 . . . . . . . . . 10  dom  F  C_
3029sseli 2935 . . . . . . . . 9  dom  F
31 rsp 2363 . . . . . . . . 9  C_  C  C_  C
3230, 31mpan9 265 . . . . . . . 8  dom  F  C_  C  C_  C
3328, 32sstrd 2949 . . . . . . 7  dom  F  C_  C  F ` 
C_  C
3433ex 108 . . . . . 6  dom  F  C_  C  F `  C_  C
3522, 34sylbir 125 . . . . 5  {  |  _V }  C_  C  F `
 C_  C
3615, 20, 35chvar 1637 . . . 4  {  |  _V }  C_  C  F `
 C_  C
373, 36vtoclga 2613 . . 3  D  {  |  _V }  C_  C  F `
 D  C_  C
3837, 21eleq2s 2129 . 2  D  dom  F  C_  C  F `  D  C_  C
3938imp 115 1  D  dom  F  C_  C  F `  D 
C_  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   wcel 1390  wral 2300   {crab 2304   _Vcvv 2551    C_ wss 2911    |-> cmpt 3809   dom cdm 4288   ` cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fv 4853
This theorem is referenced by: (None)
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