Theorem dfsucmap4 39214

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 2767	. . 3 ⊢ (𝑛 = suc 𝑚 ↔ suc 𝑚 = 𝑛)
2	1	opabbii 5172	. 2 ⊢ {⟨𝑚, 𝑛⟩ ∣ 𝑛 = suc 𝑚} = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
3		mptv 5211	. 2 ⊢ (𝑚 ∈ V ↦ suc 𝑚) = {⟨𝑚, 𝑛⟩ ∣ 𝑛 = suc 𝑚}
4		df-sucmap 39211	. 2 ⊢ SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
5	2, 3, 4	3eqtr4ri 2794	1 ⊢ SucMap = (𝑚 ∈ V ↦ suc 𝑚)

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3450  {copab 5167   ↦ cmpt 5186  suc csuc 6359   SucMap csucmap 38927

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

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-opab 5168  df-mpt 5187  df-sucmap 39211

This theorem is used by: (None)