Theorem brsucmap 38833

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 6378	. . 3 ⊢ (𝑚 = 𝑀 → suc 𝑚 = suc 𝑀)
2		id 22	. . 3 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁)
3	1, 2	eqeqan12d 2753	. 2 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → (suc 𝑚 = 𝑛 ↔ suc 𝑀 = 𝑁))
4		df-sucmap 38829	. 2 ⊢ SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
5	3, 4	brabga 5476	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑀 SucMap 𝑁 ↔ suc 𝑀 = 𝑁))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   class class class wbr 5072  suc csuc 6312   SucMap csucmap 38545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-suc 6316  df-sucmap 38829

This theorem is referenced by:  dmsucmap  38835  dfpre3  38845  sucmapsuc  38856  sucmapleftuniq  38857  exeupre  38858  sucpre  38864