ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dmmrnm Unicode version

Theorem dmmrnm 4643
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 4438 . . . . 5  |-  dom  A  =  { x  |  E. z  x A z }
21eleq2i 2154 . . . 4  |-  ( x  e.  dom  A  <->  x  e.  { x  |  E. z  x A z } )
32exbii 1541 . . 3  |-  ( E. x  x  e.  dom  A  <->  E. x  x  e.  { x  |  E. z  x A z } )
4 abid 2076 . . . 4  |-  ( x  e.  { x  |  E. z  x A z }  <->  E. z  x A z )
54exbii 1541 . . 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 4612 . . . . 5  |-  ran  A  =  { z  |  E. x  x A z }
87eleq2i 2154 . . . 4  |-  ( z  e.  ran  A  <->  z  e.  { z  |  E. x  x A z } )
98exbii 1541 . . 3  |-  ( E. z  z  e.  ran  A  <->  E. z  z  e.  { z  |  E. x  x A z } )
10 abid 2076 . . . . 5  |-  ( z  e.  { z  |  E. x  x A z }  <->  E. x  x A z )
1110exbii 1541 . . . 4  |-  ( E. z  z  e.  {
z  |  E. x  x A z }  <->  E. z E. x  x A
z )
12 excom 1599 . . . 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 2150 . . 3  |-  ( z  =  y  ->  (
z  e.  ran  A  <->  y  e.  ran  A ) )
1615cbvexv 1843 . 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 1426    e. wcel 1438   {cab 2074   class class class wbr 3837   dom cdm 4428   ran crn 4429
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-cnv 4436  df-dm 4438  df-rn 4439
This theorem is referenced by:  rnsnm  4884  nninfall  11546
  Copyright terms: Public domain W3C validator