Theorem brsucmap 39399

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 6431	. . 3 ⊢ (𝑚 = 𝑀 → suc 𝑚 = suc 𝑀)
2		id 23	. . 3 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁)
3	1, 2	eqeqan12d 2774	. 2 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → (suc 𝑚 = 𝑛 ↔ suc 𝑀 = 𝑁))
4		df-sucmap 39395	. 2 ⊢ SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
5	3, 4	brabga 5507	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑀 SucMap 𝑁 ↔ suc 𝑀 = 𝑁))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   class class class wbr 5102  suc csuc 6364   SucMap csucmap 39111

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-suc 6368  df-sucmap 39395

This theorem is used by:  dmsucmap  39401  dfpre3  39411  sucmapsuc  39422  sucmapleftuniq  39423  exeupre  39424  sucpre  39430