Theorem dmmrnm 4554

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 4355	. . . . 5 |- dom A = { x | E. z x A z }
2	1	eleq2i 2104	. . . 4 |- ( x e. dom A <-> x e. { x | E. z x A z } )
3	2	exbii 1496	. . 3 |- ( E. x x e. dom A <-> E. x x e. { x | E. z x A z } )
4		abid 2028	. . . 4 |- ( x e. { x | E. z x A z } <-> E. z x A z )
5	4	exbii 1496	. . 3 |- ( E. x x e. { x | E. z x A z } <-> E. x E. z x A z )
6	3, 5	bitri 173	. 2 |- ( E. x x e. dom A <-> E. x E. z x A z )
7		dfrn2 4523	. . . . 5 |- ran A = { z | E. x x A z }
8	7	eleq2i 2104	. . . 4 |- ( z e. ran A <-> z e. { z | E. x x A z } )
9	8	exbii 1496	. . 3 |- ( E. z z e. ran A <-> E. z z e. { z | E. x x A z } )
10		abid 2028	. . . . 5 |- ( z e. { z | E. x x A z } <-> E. x x A z )
11	10	exbii 1496	. . . 4 |- ( E. z z e. { z | E. x x A z } <-> E. z E. x x A z )
12		excom 1554	. . . 4 |- ( E. z E. x x A z <-> E. x E. z x A z )
13	11, 12	bitri 173	. . 3 |- ( E. z z e. { z | E. x x A z } <-> E. x E. z x A z )
14	9, 13	bitri 173	. 2 |- ( E. z z e. ran A <-> E. x E. z x A z )
15		eleq1 2100	. . 3 |- ( z = y -> ( z e. ran A <-> y e. ran A ) )
16	15	cbvexv 1795	. 2 |- ( E. z z e. ran A <-> E. y y e. ran A )
17	6, 14, 16	3bitr2i 197	1 |- ( E. x x e. dom A <-> E. y y e. ran A )

Syntax hints:   <-> wb 98   E.wex 1381   e. wcel 1393   {cab 2026   class class class wbr 3764   dom cdm 4345   ran crn 4346

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353  df-dm 4355  df-rn 4356

This theorem is referenced by:  rnsnm  4787