Theorem dfsucmap4 38488

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 2738	. . 3 ⊢ (𝑛 = suc 𝑚 ↔ suc 𝑚 = 𝑛)
2	1	opabbii 5156	. 2 ⊢ {⟨𝑚, 𝑛⟩ ∣ 𝑛 = suc 𝑚} = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
3		mptv 5195	. 2 ⊢ (𝑚 ∈ V ↦ suc 𝑚) = {⟨𝑚, 𝑛⟩ ∣ 𝑛 = suc 𝑚}
4		df-sucmap 38485	. 2 ⊢ SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
5	2, 3, 4	3eqtr4ri 2765	1 ⊢ SucMap = (𝑚 ∈ V ↦ suc 𝑚)

Syntax hints:   = wceq 1541  Vcvv 3436  {copab 5151   ↦ cmpt 5170  suc csuc 6308   SucMap csucmap 38227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-opab 5152  df-mpt 5171  df-sucmap 38485

This theorem is referenced by: (None)