Theorem dmmrnm 4846

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 4636	. . . . 5 |- dom A = { x | E. z x A z }
2	1	eleq2i 2244	. . . 4 |- ( x e. dom A <-> x e. { x | E. z x A z } )
3	2	exbii 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 )
5	4	exbii 1605	. . 3 |- ( E. x x e. { x | E. z x A z } <-> E. x E. z x A z )
6	3, 5	bitri 184	. 2 |- ( E. x x e. dom A <-> E. x E. z x A z )
7		dfrn2 4815	. . . . 5 |- ran A = { z | E. x x A z }
8	7	eleq2i 2244	. . . 4 |- ( z e. ran A <-> z e. { z | E. x x A z } )
9	8	exbii 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 )
11	10	exbii 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 )
13	11, 12	bitri 184	. . 3 |- ( E. z z e. { z | E. x x A z } <-> E. x E. z x A z )
14	9, 13	bitri 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 ) )
16	15	cbvexv 1918	. 2 |- ( E. z z e. ran A <-> E. y y e. ran A )
17	6, 14, 16	3bitr2i 208	1 |- ( E. x x e. dom A <-> E. y y e. ran A )

Syntax hints:   <-> wb 105   E.wex 1492   e. wcel 2148   {cab 2163   class class class wbr 4003   dom cdm 4626   ran crn 4627

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 4121  ax-pow 4174  ax-pr 4209

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 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-cnv 4634  df-dm 4636  df-rn 4637

This theorem is referenced by:  rnsnm  5095  nninfall  14640