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Theorem dmmrnm 4847
Description: A domain is inhabited if and only if the range is inhabited. (Contributed by Jim Kingdon, 15-Dec-2018.)
Assertion
Ref Expression
dmmrnm  |-  ( E. x  x  e.  dom  A  <->  E. y  y  e.  ran  A )
Distinct variable groups:    y, A    x, A

Proof of Theorem dmmrnm
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-dm 4637 . . . . 5  |-  dom  A  =  { x  |  E. z  x A z }
21eleq2i 2244 . . . 4  |-  ( x  e.  dom  A  <->  x  e.  { x  |  E. z  x A z } )
32exbii 1605 . . 3  |-  ( E. x  x  e.  dom  A  <->  E. x  x  e.  { x  |  E. z  x A z } )
4 abid 2165 . . . 4  |-  ( x  e.  { x  |  E. z  x A z }  <->  E. z  x A z )
54exbii 1605 . . 3  |-  ( E. x  x  e.  {
x  |  E. z  x A z }  <->  E. x E. z  x A
z )
63, 5bitri 184 . 2  |-  ( E. x  x  e.  dom  A  <->  E. x E. z  x A z )
7 dfrn2 4816 . . . . 5  |-  ran  A  =  { z  |  E. x  x A z }
87eleq2i 2244 . . . 4  |-  ( z  e.  ran  A  <->  z  e.  { z  |  E. x  x A z } )
98exbii 1605 . . 3  |-  ( E. z  z  e.  ran  A  <->  E. z  z  e.  { z  |  E. x  x A z } )
10 abid 2165 . . . . 5  |-  ( z  e.  { z  |  E. x  x A z }  <->  E. x  x A z )
1110exbii 1605 . . . 4  |-  ( E. z  z  e.  {
z  |  E. x  x A z }  <->  E. z E. x  x A
z )
12 excom 1664 . . . 4  |-  ( E. z E. x  x A z  <->  E. x E. z  x A
z )
1311, 12bitri 184 . . 3  |-  ( E. z  z  e.  {
z  |  E. x  x A z }  <->  E. x E. z  x A
z )
149, 13bitri 184 . 2  |-  ( E. z  z  e.  ran  A  <->  E. x E. z  x A z )
15 eleq1 2240 . . 3  |-  ( z  =  y  ->  (
z  e.  ran  A  <->  y  e.  ran  A ) )
1615cbvexv 1918 . 2  |-  ( E. z  z  e.  ran  A  <->  E. y  y  e.  ran  A )
176, 14, 163bitr2i 208 1  |-  ( E. x  x  e.  dom  A  <->  E. y  y  e.  ran  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 105   E.wex 1492    e. wcel 2148   {cab 2163   class class class wbr 4004   dom cdm 4627   ran crn 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-cnv 4635  df-dm 4637  df-rn 4638
This theorem is referenced by:  rnsnm  5096  nninfall  14761
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