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Theorem dmv 4551
 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 2965 . 2 dom V ⊆ V
2 dmi 4550 . . 3 dom I = V
3 ssv 2965 . . . 4 I ⊆ V
4 dmss 4534 . . . 4 ( I ⊆ V → dom I ⊆ dom V)
53, 4ax-mp 7 . . 3 dom I ⊆ dom V
62, 5eqsstr3i 2976 . 2 V ⊆ dom V
71, 6eqssi 2961 1 dom V = V
 Colors of variables: wff set class Syntax hints:   = wceq 1243  Vcvv 2557   ⊆ wss 2917   I cid 4025  dom cdm 4345 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-ral 2311  df-rex 2312  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-id 4030  df-xp 4351  df-rel 4352  df-dm 4355 This theorem is referenced by: (None)
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