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Theorem dmmrnm 4613
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 4411 . . . . 5  |-  dom  A  =  { x  |  E. z  x A z }
21eleq2i 2149 . . . 4  |-  ( x  e.  dom  A  <->  x  e.  { x  |  E. z  x A z } )
32exbii 1537 . . 3  |-  ( E. x  x  e.  dom  A  <->  E. x  x  e.  { x  |  E. z  x A z } )
4 abid 2071 . . . 4  |-  ( x  e.  { x  |  E. z  x A z }  <->  E. z  x A z )
54exbii 1537 . . 3  |-  ( E. x  x  e.  {
x  |  E. z  x A z }  <->  E. x E. z  x A
z )
63, 5bitri 182 . 2  |-  ( E. x  x  e.  dom  A  <->  E. x E. z  x A z )
7 dfrn2 4582 . . . . 5  |-  ran  A  =  { z  |  E. x  x A z }
87eleq2i 2149 . . . 4  |-  ( z  e.  ran  A  <->  z  e.  { z  |  E. x  x A z } )
98exbii 1537 . . 3  |-  ( E. z  z  e.  ran  A  <->  E. z  z  e.  { z  |  E. x  x A z } )
10 abid 2071 . . . . 5  |-  ( z  e.  { z  |  E. x  x A z }  <->  E. x  x A z )
1110exbii 1537 . . . 4  |-  ( E. z  z  e.  {
z  |  E. x  x A z }  <->  E. z E. x  x A
z )
12 excom 1595 . . . 4  |-  ( E. z E. x  x A z  <->  E. x E. z  x A
z )
1311, 12bitri 182 . . 3  |-  ( E. z  z  e.  {
z  |  E. x  x A z }  <->  E. x E. z  x A
z )
149, 13bitri 182 . 2  |-  ( E. z  z  e.  ran  A  <->  E. x E. z  x A z )
15 eleq1 2145 . . 3  |-  ( z  =  y  ->  (
z  e.  ran  A  <->  y  e.  ran  A ) )
1615cbvexv 1838 . 2  |-  ( E. z  z  e.  ran  A  <->  E. y  y  e.  ran  A )
176, 14, 163bitr2i 206 1  |-  ( E. x  x  e.  dom  A  <->  E. y  y  e.  ran  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   E.wex 1422    e. wcel 1434   {cab 2069   class class class wbr 3811   dom cdm 4401   ran crn 4402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-cnv 4409  df-dm 4411  df-rn 4412
This theorem is referenced by:  rnsnm  4851  nninfall  11241
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