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Theorem fvmptss2 5190
Description: A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2019.)
Hypotheses
Ref Expression
fvmptss2.1  D  C
fvmptss2.2  F  |->
Assertion
Ref Expression
fvmptss2  F `
 D  C_  C
Distinct variable groups:   ,   , C   , D
Allowed substitution hints:   ()    F()

Proof of Theorem fvmptss2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fvss 5132 . 2  D F  C_  C  F `  D  C_  C
2 fvmptss2.2 . . . . . 6  F  |->
32funmpt2 4882 . . . . 5  Fun  F
4 funrel 4862 . . . . 5  Fun 
F  Rel  F
53, 4ax-mp 7 . . . 4  Rel  F
65brrelexi 4327 . . 3  D F  D  _V
7 nfcv 2175 . . . 4  F/_ D
8 nfmpt1 3841 . . . . . . 7  F/_  |->
92, 8nfcxfr 2172 . . . . . 6  F/_ F
10 nfcv 2175 . . . . . 6  F/_
117, 9, 10nfbr 3799 . . . . 5  F/  D F
12 nfv 1418 . . . . 5  F/  C_  C
1311, 12nfim 1461 . . . 4  F/ D F  C_  C
14 breq1 3758 . . . . 5  D  F  D F
15 fvmptss2.1 . . . . . 6  D  C
1615sseq2d 2967 . . . . 5  D  C_  C_  C
1714, 16imbi12d 223 . . . 4  D  F  C_  D F  C_  C
18 df-br 3756 . . . . 5  F  <. ,  >.  F
19 opabid 3985 . . . . . . 7  <. ,  >.  { <. ,  >.  |  }
20 eqimss 2991 . . . . . . . 8  C_
2120adantl 262 . . . . . . 7  C_
2219, 21sylbi 114 . . . . . 6  <. ,  >.  { <. ,  >.  |  }  C_
23 df-mpt 3811 . . . . . . 7  |->  { <. ,  >.  |  }
242, 23eqtri 2057 . . . . . 6  F  { <. , 
>.  |  }
2522, 24eleq2s 2129 . . . . 5  <. ,  >.  F  C_
2618, 25sylbi 114 . . . 4  F  C_
277, 13, 17, 26vtoclgf 2606 . . 3  D  _V  D F  C_  C
286, 27mpcom 32 . 2  D F  C_  C
291, 28mpg 1337 1  F `
 D  C_  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   wcel 1390   _Vcvv 2551    C_ wss 2911   <.cop 3370   class class class wbr 3755   {copab 3808    |-> cmpt 3809   Rel wrel 4293   Fun wfun 4839   ` 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-v 2553  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-iota 4810  df-fun 4847  df-fv 4853
This theorem is referenced by:  mptfvex  5199
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