Theorem dmsucmap 39666

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 3937	. 2 ⊢ dom SucMap ⊆ V
2		sucexg 7853	. . . . . . 7 ⊢ (𝑚 ∈ V → suc 𝑚 ∈ V)
3	2	elv 3459	. . . . . 6 ⊢ suc 𝑚 ∈ V
4	3	isseti 3472	. . . . 5 ⊢ ∃𝑛 𝑛 = suc 𝑚
5		brsucmap 39664	. . . . . . . 8 ⊢ ((𝑚 ∈ V ∧ 𝑛 ∈ V) → (𝑚 SucMap 𝑛 ↔ suc 𝑚 = 𝑛))
6	5	el2v 3461	. . . . . . 7 ⊢ (𝑚 SucMap 𝑛 ↔ suc 𝑚 = 𝑛)
7		eqcom 2770	. . . . . . 7 ⊢ (suc 𝑚 = 𝑛 ↔ 𝑛 = suc 𝑚)
8	6, 7	bitri 278	. . . . . 6 ⊢ (𝑚 SucMap 𝑛 ↔ 𝑛 = suc 𝑚)
9	8	exbii 1881	. . . . 5 ⊢ (∃𝑛 𝑚 SucMap 𝑛 ↔ ∃𝑛 𝑛 = suc 𝑚)
10	4, 9	mpbir 234	. . . 4 ⊢ ∃𝑛 𝑚 SucMap 𝑛
11	10	rgenw 3084	. . 3 ⊢ ∀𝑚 ∈ V ∃𝑛 𝑚 SucMap 𝑛
12		ssdmral 39577	. . 3 ⊢ (V ⊆ dom SucMap ↔ ∀𝑚 ∈ V ∃𝑛 𝑚 SucMap 𝑛)
13	11, 12	mpbir 234	. 2 ⊢ V ⊆ dom SucMap
14	1, 13	eqssi 3929	1 ⊢ dom SucMap = V

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   ⊆ wss 3900   class class class wbr 5108  dom cdm 5658  suc csuc 6390   SucMap csucmap 39376

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-dm 5668  df-suc 6394  df-sucmap 39660

This theorem is used by:  dfpre  39674