Theorem brsucmap 39279

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 6428	. . 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 39275	. 2 ⊢ SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
5	3, 4	brabga 5512	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑀 SucMap 𝑁 ↔ suc 𝑀 = 𝑁))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   class class class wbr 5103  suc csuc 6361   SucMap csucmap 38991

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-suc 6365  df-sucmap 39275

This theorem is used by:  dmsucmap  39281  dfpre3  39291  sucmapsuc  39302  sucmapleftuniq  39303  exeupre  39304  sucpre  39310