Theorem dmsucmap 38581

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 3956	. 2 ⊢ dom SucMap ⊆ V
2		sucexg 7748	. . . . . . 7 ⊢ (𝑚 ∈ V → suc 𝑚 ∈ V)
3	2	elv 3443	. . . . . 6 ⊢ suc 𝑚 ∈ V
4	3	isseti 3456	. . . . 5 ⊢ ∃𝑛 𝑛 = suc 𝑚
5		brsucmap 38579	. . . . . . . 8 ⊢ ((𝑚 ∈ V ∧ 𝑛 ∈ V) → (𝑚 SucMap 𝑛 ↔ suc 𝑚 = 𝑛))
6	5	el2v 3445	. . . . . . 7 ⊢ (𝑚 SucMap 𝑛 ↔ suc 𝑚 = 𝑛)
7		eqcom 2741	. . . . . . 7 ⊢ (suc 𝑚 = 𝑛 ↔ 𝑛 = suc 𝑚)
8	6, 7	bitri 275	. . . . . 6 ⊢ (𝑚 SucMap 𝑛 ↔ 𝑛 = suc 𝑚)
9	8	exbii 1849	. . . . 5 ⊢ (∃𝑛 𝑚 SucMap 𝑛 ↔ ∃𝑛 𝑛 = suc 𝑚)
10	4, 9	mpbir 231	. . . 4 ⊢ ∃𝑛 𝑚 SucMap 𝑛
11	10	rgenw 3053	. . 3 ⊢ ∀𝑚 ∈ V ∃𝑛 𝑚 SucMap 𝑛
12		ssdmral 38503	. . 3 ⊢ (V ⊆ dom SucMap ↔ ∀𝑚 ∈ V ∃𝑛 𝑚 SucMap 𝑛)
13	11, 12	mpbir 231	. 2 ⊢ V ⊆ dom SucMap
14	1, 13	eqssi 3948	1 ⊢ dom SucMap = V

Syntax hints:   ↔ wb 206   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3049  Vcvv 3438   ⊆ wss 3899   class class class wbr 5096  dom cdm 5622  suc csuc 6317   SucMap csucmap 38317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-dm 5632  df-suc 6321  df-sucmap 38575

This theorem is referenced by:  dfpre  38589