Theorem dmsucmap 38835

Description: The domain of the successor map is the universe. (Contributed by Peter Mazsa, 7-Jan-2026.)

Assertion

Ref	Expression
dmsucmap	⊢ dom SucMap = V

Proof of Theorem dmsucmap

Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssv 3939	. 2 ⊢ dom SucMap ⊆ V
2		sucexg 7748	. . . . . . 7 ⊢ (𝑚 ∈ V → suc 𝑚 ∈ V)
3	2	elv 3436	. . . . . 6 ⊢ suc 𝑚 ∈ V
4	3	isseti 3449	. . . . 5 ⊢ ∃𝑛 𝑛 = suc 𝑚
5		brsucmap 38833	. . . . . . . 8 ⊢ ((𝑚 ∈ V ∧ 𝑛 ∈ V) → (𝑚 SucMap 𝑛 ↔ suc 𝑚 = 𝑛))
6	5	el2v 3438	. . . . . . 7 ⊢ (𝑚 SucMap 𝑛 ↔ suc 𝑚 = 𝑛)
7		eqcom 2746	. . . . . . 7 ⊢ (suc 𝑚 = 𝑛 ↔ 𝑛 = suc 𝑚)
8	6, 7	bitri 276	. . . . . 6 ⊢ (𝑚 SucMap 𝑛 ↔ 𝑛 = suc 𝑚)
9	8	exbii 1855	. . . . 5 ⊢ (∃𝑛 𝑚 SucMap 𝑛 ↔ ∃𝑛 𝑛 = suc 𝑚)
10	4, 9	mpbir 232	. . . 4 ⊢ ∃𝑛 𝑚 SucMap 𝑛
11	10	rgenw 3057	. . 3 ⊢ ∀𝑚 ∈ V ∃𝑛 𝑚 SucMap 𝑛
12		ssdmral 38746	. . 3 ⊢ (V ⊆ dom SucMap ↔ ∀𝑚 ∈ V ∃𝑛 𝑚 SucMap 𝑛)
13	11, 12	mpbir 232	. 2 ⊢ V ⊆ dom SucMap
14	1, 13	eqssi 3931	1 ⊢ dom SucMap = V

Syntax hints:   ↔ wb 207   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∀wral 3053  Vcvv 3431   ⊆ wss 3883   class class class wbr 5072  dom cdm 5618  suc csuc 6312   SucMap csucmap 38545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-dm 5628  df-suc 6316  df-sucmap 38829

This theorem is referenced by:  dfpre  38843