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Theorem dmmrnm 4762
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 4553 . . . . 5  |-  dom  A  =  { x  |  E. z  x A z }
21eleq2i 2207 . . . 4  |-  ( x  e.  dom  A  <->  x  e.  { x  |  E. z  x A z } )
32exbii 1585 . . 3  |-  ( E. x  x  e.  dom  A  <->  E. x  x  e.  { x  |  E. z  x A z } )
4 abid 2128 . . . 4  |-  ( x  e.  { x  |  E. z  x A z }  <->  E. z  x A z )
54exbii 1585 . . 3  |-  ( E. x  x  e.  {
x  |  E. z  x A z }  <->  E. x E. z  x A
z )
63, 5bitri 183 . 2  |-  ( E. x  x  e.  dom  A  <->  E. x E. z  x A z )
7 dfrn2 4731 . . . . 5  |-  ran  A  =  { z  |  E. x  x A z }
87eleq2i 2207 . . . 4  |-  ( z  e.  ran  A  <->  z  e.  { z  |  E. x  x A z } )
98exbii 1585 . . 3  |-  ( E. z  z  e.  ran  A  <->  E. z  z  e.  { z  |  E. x  x A z } )
10 abid 2128 . . . . 5  |-  ( z  e.  { z  |  E. x  x A z }  <->  E. x  x A z )
1110exbii 1585 . . . 4  |-  ( E. z  z  e.  {
z  |  E. x  x A z }  <->  E. z E. x  x A
z )
12 excom 1643 . . . 4  |-  ( E. z E. x  x A z  <->  E. x E. z  x A
z )
1311, 12bitri 183 . . 3  |-  ( E. z  z  e.  {
z  |  E. x  x A z }  <->  E. x E. z  x A
z )
149, 13bitri 183 . 2  |-  ( E. z  z  e.  ran  A  <->  E. x E. z  x A z )
15 eleq1 2203 . . 3  |-  ( z  =  y  ->  (
z  e.  ran  A  <->  y  e.  ran  A ) )
1615cbvexv 1891 . 2  |-  ( E. z  z  e.  ran  A  <->  E. y  y  e.  ran  A )
176, 14, 163bitr2i 207 1  |-  ( E. x  x  e.  dom  A  <->  E. y  y  e.  ran  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   E.wex 1469    e. wcel 1481   {cab 2126   class class class wbr 3933   dom cdm 4543   ran crn 4544
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2689  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-br 3934  df-opab 3994  df-cnv 4551  df-dm 4553  df-rn 4554
This theorem is referenced by:  rnsnm  5009  nninfall  13362
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