Theorem dmmrnm 4916

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 4703	. . . . 5 |- dom A = { x | E. z x A z }
2	1	eleq2i 2274	. . . 4 |- ( x e. dom A <-> x e. { x | E. z x A z } )
3	2	exbii 1629	. . 3 |- ( E. x x e. dom A <-> E. x x e. { x | E. z x A z } )
4		abid 2195	. . . 4 |- ( x e. { x | E. z x A z } <-> E. z x A z )
5	4	exbii 1629	. . 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 4884	. . . . 5 |- ran A = { z | E. x x A z }
8	7	eleq2i 2274	. . . 4 |- ( z e. ran A <-> z e. { z | E. x x A z } )
9	8	exbii 1629	. . 3 |- ( E. z z e. ran A <-> E. z z e. { z | E. x x A z } )
10		abid 2195	. . . . 5 |- ( z e. { z | E. x x A z } <-> E. x x A z )
11	10	exbii 1629	. . . 4 |- ( E. z z e. { z | E. x x A z } <-> E. z E. x x A z )
12		excom 1688	. . . 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 2270	. . 3 |- ( z = y -> ( z e. ran A <-> y e. ran A ) )
16	15	cbvexv 1943	. 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 1516   e. wcel 2178   {cab 2193   class class class wbr 4059   dom cdm 4693   ran crn 4694

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-cnv 4701  df-dm 4703  df-rn 4704

This theorem is referenced by:  rnsnm  5168  ghmrn  13708  nninfall  16148