Theorem dfsucmap4 38832

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 2746	. . 3 ⊢ (𝑛 = suc 𝑚 ↔ suc 𝑚 = 𝑛)
2	1	opabbii 5139	. 2 ⊢ {⟨𝑚, 𝑛⟩ ∣ 𝑛 = suc 𝑚} = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
3		mptv 5178	. 2 ⊢ (𝑚 ∈ V ↦ suc 𝑚) = {⟨𝑚, 𝑛⟩ ∣ 𝑛 = suc 𝑚}
4		df-sucmap 38829	. 2 ⊢ SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
5	2, 3, 4	3eqtr4ri 2773	1 ⊢ SucMap = (𝑚 ∈ V ↦ suc 𝑚)

Syntax hints:   = wceq 1547  Vcvv 3431  {copab 5134   ↦ cmpt 5153  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

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-opab 5135  df-mpt 5154  df-sucmap 38829

This theorem is referenced by: (None)