Theorem brsucmap 39143

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 6429	. . 3 ⊢ (𝑚 = 𝑀 → suc 𝑚 = suc 𝑀)
2		id 23	. . 3 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁)
3	1, 2	eqeqan12d 2777	. 2 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → (suc 𝑚 = 𝑛 ↔ suc 𝑀 = 𝑁))
4		df-sucmap 39139	. 2 ⊢ SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
5	3, 4	brabga 5518	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑀 SucMap 𝑁 ↔ suc 𝑀 = 𝑁))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   class class class wbr 5109  suc csuc 6362   SucMap csucmap 38855

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-suc 6366  df-sucmap 39139

This theorem is used by:  dmsucmap  39145  dfpre3  39155  sucmapsuc  39166  sucmapleftuniq  39167  exeupre  39168  sucpre  39174