Theorem dmv 5497

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 3767	. 2 ⊢ dom V ⊆ V
2		dmi 5496	. . 3 ⊢ dom I = V
3		ssv 3767	. . . 4 ⊢ I ⊆ V
4		dmss 5479	. . . 4 ⊢ ( I ⊆ V → dom I ⊆ dom V)
5	3, 4	ax-mp 5	. . 3 ⊢ dom I ⊆ dom V
6	2, 5	eqsstr3i 3778	. 2 ⊢ V ⊆ dom V
7	1, 6	eqssi 3761	1 ⊢ dom V = V

Syntax hints:   = wceq 1632  Vcvv 3341   ⊆ wss 3716   I cid 5174  dom cdm 5267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-id 5175  df-xp 5273  df-rel 5274  df-dm 5277

This theorem is referenced by: (None)