Theorem brsucmap 38579

Description: Binary relation form of the successor map, general version. (Contributed by Peter Mazsa, 6-Jan-2026.)

Assertion

Ref	Expression
brsucmap	⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑀 SucMap 𝑁 ↔ suc 𝑀 = 𝑁))

Proof of Theorem brsucmap

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

Step	Hyp	Ref	Expression
1		suceq 6383	. . 3 ⊢ (𝑚 = 𝑀 → suc 𝑚 = suc 𝑀)
2		id 22	. . 3 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁)
3	1, 2	eqeqan12d 2748	. 2 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → (suc 𝑚 = 𝑛 ↔ suc 𝑀 = 𝑁))
4		df-sucmap 38575	. 2 ⊢ SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
5	3, 4	brabga 5480	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑀 SucMap 𝑁 ↔ suc 𝑀 = 𝑁))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   class class class wbr 5096  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

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-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-suc 6321  df-sucmap 38575

This theorem is referenced by:  dmsucmap  38581  dfpre3  38591  sucmapsuc  38601  sucmapleftuniq  38602  exeupre  38603  sucpre  38609