Theorem dfsucmap4 39537

Description: Alternate definition of the successor map. (Contributed by Peter Mazsa, 28-Jan-2026.)

Assertion

Ref	Expression
dfsucmap4	⊢ SucMap = (𝑚 ∈ V ↦ suc 𝑚)

Proof of Theorem dfsucmap4

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqcom 2770	. . 3 ⊢ (𝑛 = suc 𝑚 ↔ suc 𝑚 = 𝑛)
2	1	opabbii 5174	. 2 ⊢ {⟨𝑚, 𝑛⟩ ∣ 𝑛 = suc 𝑚} = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
3		mptv 5213	. 2 ⊢ (𝑚 ∈ V ↦ suc 𝑚) = {⟨𝑚, 𝑛⟩ ∣ 𝑛 = suc 𝑚}
4		df-sucmap 39534	. 2 ⊢ SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
5	2, 3, 4	3eqtr4ri 2797	1 ⊢ SucMap = (𝑚 ∈ V ↦ suc 𝑚)

This proof depends on syntax axioms:   = wceq 1569  Vcvv 3453  {copab 5169   ↦ cmpt 5188  suc csuc 6370   SucMap csucmap 39250

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3455  df-opab 5170  df-mpt 5189  df-sucmap 39534

This theorem is used by: (None)