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Theorem fvmptss2 5193
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 5135 . 2  D F  C_  C  F `  D  C_  C
2 fvmptss2.2 . . . . . 6  F  |->
32funmpt2 4885 . . . . 5  Fun  F
4 funrel 4865 . . . . 5  Fun 
F  Rel  F
53, 4ax-mp 7 . . . 4  Rel  F
65brrelexi 4330 . . 3  D F  D  _V
7 nfcv 2178 . . . 4  F/_ D
8 nfmpt1 3844 . . . . . . 7  F/_  |->
92, 8nfcxfr 2175 . . . . . 6  F/_ F
10 nfcv 2178 . . . . . 6  F/_
117, 9, 10nfbr 3802 . . . . 5  F/  D F
12 nfv 1421 . . . . 5  F/  C_  C
1311, 12nfim 1464 . . . 4  F/ D F  C_  C
14 breq1 3761 . . . . 5  D  F  D F
15 fvmptss2.1 . . . . . 6  D  C
1615sseq2d 2970 . . . . 5  D  C_  C_  C
1714, 16imbi12d 223 . . . 4  D  F  C_  D F  C_  C
18 df-br 3759 . . . . 5  F  <. ,  >.  F
19 opabid 3988 . . . . . . 7  <. ,  >.  { <. ,  >.  |  }
20 eqimss 2994 . . . . . . . 8  C_
2120adantl 262 . . . . . . 7  C_
2219, 21sylbi 114 . . . . . 6  <. ,  >.  { <. ,  >.  |  }  C_
23 df-mpt 3814 . . . . . . 7  |->  { <. ,  >.  |  }
242, 23eqtri 2060 . . . . . 6  F  { <. , 
>.  |  }
2522, 24eleq2s 2132 . . . . 5  <. ,  >.  F  C_
2618, 25sylbi 114 . . . 4  F  C_
277, 13, 17, 26vtoclgf 2609 . . 3  D  _V  D F  C_  C
286, 27mpcom 32 . 2  D F  C_  C
291, 28mpg 1340 1  F `
 D  C_  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1243   wcel 1393   _Vcvv 2554    C_ wss 2914   <.cop 3373   class class class wbr 3758   {copab 3811    |-> cmpt 3812   Rel wrel 4296   Fun wfun 4842   ` cfv 4848
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3869  ax-pow 3921  ax-pr 3938
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-pw 3356  df-sn 3376  df-pr 3377  df-op 3379  df-uni 3575  df-br 3759  df-opab 3813  df-mpt 3814  df-id 4024  df-xp 4297  df-rel 4298  df-cnv 4299  df-co 4300  df-iota 4813  df-fun 4850  df-fv 4856
This theorem is referenced by:  mptfvex  5202
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