Theorem dmsucmap 39175

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 3962	. 2 ⊢ dom SucMap ⊆ V
2		sucexg 7810	. . . . . . 7 ⊢ (𝑚 ∈ V → suc 𝑚 ∈ V)
3	2	elv 3462	. . . . . 6 ⊢ suc 𝑚 ∈ V
4	3	isseti 3475	. . . . 5 ⊢ ∃𝑛 𝑛 = suc 𝑚
5		brsucmap 39173	. . . . . . . 8 ⊢ ((𝑚 ∈ V ∧ 𝑛 ∈ V) → (𝑚 SucMap 𝑛 ↔ suc 𝑚 = 𝑛))
6	5	el2v 3464	. . . . . . 7 ⊢ (𝑚 SucMap 𝑛 ↔ suc 𝑚 = 𝑛)
7		eqcom 2772	. . . . . . 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 3085	. . 3 ⊢ ∀𝑚 ∈ V ∃𝑛 𝑚 SucMap 𝑛
12		ssdmral 39086	. . 3 ⊢ (V ⊆ dom SucMap ↔ ∀𝑚 ∈ V ∃𝑛 𝑚 SucMap 𝑛)
13	11, 12	mpbir 234	. 2 ⊢ V ⊆ dom SucMap
14	1, 13	eqssi 3954	1 ⊢ dom SucMap = V

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∀wral 3081  Vcvv 3457   ⊆ wss 3906   class class class wbr 5111  dom cdm 5663  suc csuc 6366   SucMap csucmap 38885

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406  ax-un 7742

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 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-dm 5673  df-suc 6370  df-sucmap 39169

This theorem is used by:  dfpre  39183