Theorem dfsucmap4 39003

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 2776	. . 3 ⊢ (𝑛 = suc 𝑚 ↔ suc 𝑚 = 𝑛)
2	1	opabbii 5182	. 2 ⊢ {⟨𝑚, 𝑛⟩ ∣ 𝑛 = suc 𝑚} = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
3		mptv 5221	. 2 ⊢ (𝑚 ∈ V ↦ suc 𝑚) = {⟨𝑚, 𝑛⟩ ∣ 𝑛 = suc 𝑚}
4		df-sucmap 39000	. 2 ⊢ SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
5	2, 3, 4	3eqtr4ri 2803	1 ⊢ SucMap = (𝑚 ∈ V ↦ suc 𝑚)

Syntax hints:   = wceq 1567  Vcvv 3463  {copab 5177   ↦ cmpt 5196  suc csuc 6363   SucMap csucmap 38716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-opab 5178  df-mpt 5197  df-sucmap 39000

This theorem is referenced by: (None)