MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmv Structured version   Visualization version   GIF version

Theorem dmv 5373
Description: The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.)
Assertion
Ref Expression
dmv dom V = V

Proof of Theorem dmv
StepHypRef Expression
1 ssv 3658 . 2 dom V ⊆ V
2 dmi 5372 . . 3 dom I = V
3 ssv 3658 . . . 4 I ⊆ V
4 dmss 5355 . . . 4 ( I ⊆ V → dom I ⊆ dom V)
53, 4ax-mp 5 . . 3 dom I ⊆ dom V
62, 5eqsstr3i 3669 . 2 V ⊆ dom V
71, 6eqssi 3652 1 dom V = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  Vcvv 3231  wss 3607   I cid 5052  dom cdm 5143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-dm 5153
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator