Theorem dmsucmap 38491

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 3954	. 2 ⊢ dom SucMap ⊆ V
2		sucexg 7738	. . . . . . 7 ⊢ (𝑚 ∈ V → suc 𝑚 ∈ V)
3	2	elv 3441	. . . . . 6 ⊢ suc 𝑚 ∈ V
4	3	isseti 3454	. . . . 5 ⊢ ∃𝑛 𝑛 = suc 𝑚
5		brsucmap 38489	. . . . . . . 8 ⊢ ((𝑚 ∈ V ∧ 𝑛 ∈ V) → (𝑚 SucMap 𝑛 ↔ suc 𝑚 = 𝑛))
6	5	el2v 3443	. . . . . . 7 ⊢ (𝑚 SucMap 𝑛 ↔ suc 𝑚 = 𝑛)
7		eqcom 2738	. . . . . . 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 3051	. . 3 ⊢ ∀𝑚 ∈ V ∃𝑛 𝑚 SucMap 𝑛
12		ssdmral 38413	. . 3 ⊢ (V ⊆ dom SucMap ↔ ∀𝑚 ∈ V ∃𝑛 𝑚 SucMap 𝑛)
13	11, 12	mpbir 231	. 2 ⊢ V ⊆ dom SucMap
14	1, 13	eqssi 3946	1 ⊢ dom SucMap = V

Syntax hints:   ↔ wb 206   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ⊆ wss 3897   class class class wbr 5089  dom cdm 5614  suc csuc 6308   SucMap csucmap 38227

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

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 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-dm 5624  df-suc 6312  df-sucmap 38485

This theorem is referenced by:  dfpre  38499