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

Step	Hyp	Ref	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)
5	3, 4	ax-mp 5	. . 3 ⊢ dom I ⊆ dom V
6	2, 5	eqsstr3i 3669	. 2 ⊢ V ⊆ dom V
7	1, 6	eqssi 3652	1 ⊢ dom V = V

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)