Theorem dmmrnm 4817

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.

Step	Hyp	Ref	Expression
1		df-dm 4608	. . . . 5 |- dom A = { x | E. z x A z }
2	1	eleq2i 2231	. . . 4 |- ( x e. dom A <-> x e. { x | E. z x A z } )
3	2	exbii 1592	. . 3 |- ( E. x x e. dom A <-> E. x x e. { x | E. z x A z } )
4		abid 2152	. . . 4 |- ( x e. { x | E. z x A z } <-> E. z x A z )
5	4	exbii 1592	. . 3 |- ( E. x x e. { x | E. z x A z } <-> E. x E. z x A z )
6	3, 5	bitri 183	. 2 |- ( E. x x e. dom A <-> E. x E. z x A z )
7		dfrn2 4786	. . . . 5 |- ran A = { z | E. x x A z }
8	7	eleq2i 2231	. . . 4 |- ( z e. ran A <-> z e. { z | E. x x A z } )
9	8	exbii 1592	. . 3 |- ( E. z z e. ran A <-> E. z z e. { z | E. x x A z } )
10		abid 2152	. . . . 5 |- ( z e. { z | E. x x A z } <-> E. x x A z )
11	10	exbii 1592	. . . 4 |- ( E. z z e. { z | E. x x A z } <-> E. z E. x x A z )
12		excom 1651	. . . 4 |- ( E. z E. x x A z <-> E. x E. z x A z )
13	11, 12	bitri 183	. . 3 |- ( E. z z e. { z | E. x x A z } <-> E. x E. z x A z )
14	9, 13	bitri 183	. 2 |- ( E. z z e. ran A <-> E. x E. z x A z )
15		eleq1 2227	. . 3 |- ( z = y -> ( z e. ran A <-> y e. ran A ) )
16	15	cbvexv 1905	. 2 |- ( E. z z e. ran A <-> E. y y e. ran A )
17	6, 14, 16	3bitr2i 207	1 |- ( E. x x e. dom A <-> E. y y e. ran A )

Syntax hints:   <-> wb 104   E.wex 1479   e. wcel 2135   {cab 2150   class class class wbr 3976   dom cdm 4598   ran crn 4599

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-br 3977  df-opab 4038  df-cnv 4606  df-dm 4608  df-rn 4609

This theorem is referenced by:  rnsnm  5064  nninfall  13723