Theorem dmmrnm 4830

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 4621	. . . . 5 |- dom A = { x | E. z x A z }
2	1	eleq2i 2237	. . . 4 |- ( x e. dom A <-> x e. { x | E. z x A z } )
3	2	exbii 1598	. . 3 |- ( E. x x e. dom A <-> E. x x e. { x | E. z x A z } )
4		abid 2158	. . . 4 |- ( x e. { x | E. z x A z } <-> E. z x A z )
5	4	exbii 1598	. . 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 4799	. . . . 5 |- ran A = { z | E. x x A z }
8	7	eleq2i 2237	. . . 4 |- ( z e. ran A <-> z e. { z | E. x x A z } )
9	8	exbii 1598	. . 3 |- ( E. z z e. ran A <-> E. z z e. { z | E. x x A z } )
10		abid 2158	. . . . 5 |- ( z e. { z | E. x x A z } <-> E. x x A z )
11	10	exbii 1598	. . . 4 |- ( E. z z e. { z | E. x x A z } <-> E. z E. x x A z )
12		excom 1657	. . . 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 2233	. . 3 |- ( z = y -> ( z e. ran A <-> y e. ran A ) )
16	15	cbvexv 1911	. 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 1485   e. wcel 2141   {cab 2156   class class class wbr 3989   dom cdm 4611   ran crn 4612

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-cnv 4619  df-dm 4621  df-rn 4622

This theorem is referenced by:  rnsnm  5077  nninfall  14042