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Theorem dmmptg 5006
Description: The domain of the mapping operation is the stated domain, if the function value is always a set. (Contributed by Mario Carneiro, 9-Feb-2013.) (Revised by Mario Carneiro, 14-Sep-2013.)
Assertion
Ref Expression
dmmptg  |-  ( A. x  e.  A  B  e.  V  ->  dom  (
x  e.  A  |->  B )  =  A )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    V( x)

Proof of Theorem dmmptg
StepHypRef Expression
1 elex 2671 . . . 4  |-  ( B  e.  V  ->  B  e.  _V )
21ralimi 2472 . . 3  |-  ( A. x  e.  A  B  e.  V  ->  A. x  e.  A  B  e.  _V )
3 rabid2 2584 . . 3  |-  ( A  =  { x  e.  A  |  B  e. 
_V }  <->  A. x  e.  A  B  e.  _V )
42, 3sylibr 133 . 2  |-  ( A. x  e.  A  B  e.  V  ->  A  =  { x  e.  A  |  B  e.  _V } )
5 eqid 2117 . . 3  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
65dmmpt 5004 . 2  |-  dom  (
x  e.  A  |->  B )  =  { x  e.  A  |  B  e.  _V }
74, 6syl6reqr 2169 1  |-  ( A. x  e.  A  B  e.  V  ->  dom  (
x  e.  A  |->  B )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465   A.wral 2393   {crab 2397   _Vcvv 2660    |-> cmpt 3959   dom cdm 4509
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-mpt 3961  df-xp 4515  df-rel 4516  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522
This theorem is referenced by:  resfunexg  5609  rdgtfr  6239  rdgruledefgg  6240  negfi  10967  limccnp2lem  12741  dvmptclx  12776  dvmptaddx  12777  dvmptmulx  12778
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