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Theorem dmmrnm 4762
 Description: A domain is inhabited if and only if the range is inhabited. (Contributed by Jim Kingdon, 15-Dec-2018.)
Assertion
Ref Expression
dmmrnm
Distinct variable groups:   ,   ,

Proof of Theorem dmmrnm
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-dm 4553 . . . . 5
21eleq2i 2207 . . . 4
32exbii 1585 . . 3
4 abid 2128 . . . 4
54exbii 1585 . . 3
63, 5bitri 183 . 2
7 dfrn2 4731 . . . . 5
87eleq2i 2207 . . . 4
98exbii 1585 . . 3
10 abid 2128 . . . . 5
1110exbii 1585 . . . 4
12 excom 1643 . . . 4
1311, 12bitri 183 . . 3
149, 13bitri 183 . 2
15 eleq1 2203 . . 3
1615cbvexv 1891 . 2
176, 14, 163bitr2i 207 1
 Colors of variables: wff set class Syntax hints:   wb 104  wex 1469   wcel 1481  cab 2126   class class class wbr 3933   cdm 4543   crn 4544 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2689  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-br 3934  df-opab 3994  df-cnv 4551  df-dm 4553  df-rn 4554 This theorem is referenced by:  rnsnm  5009  nninfall  13362
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