Theorem dfsuccl2 39008

Description: Alternate definition of the class of all successors. (Contributed by Peter Mazsa, 29-Jan-2026.)

Assertion

Ref	Expression
dfsuccl2	⊢ Suc = {𝑛 ∣ ∃𝑚 suc 𝑚 = 𝑛}

Distinct variable group:   𝑚,𝑛

Proof of Theorem dfsuccl2

Step	Hyp	Ref	Expression
1		df-succl 39007	. 2 ⊢ Suc = ran SucMap
2		df-sucmap 39000	. . 3 ⊢ SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
3	2	rneqi 5928	. 2 ⊢ ran SucMap = ran {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
4		rnopab 5945	. 2 ⊢ ran {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛} = {𝑛 ∣ ∃𝑚 suc 𝑚 = 𝑛}
5	1, 3, 4	3eqtri 2796	1 ⊢ Suc = {𝑛 ∣ ∃𝑚 suc 𝑚 = 𝑛}

Syntax hints:   = wceq 1567  ∃wex 1806  {cab 2747  {copab 5177  ran crn 5663  suc csuc 6363   SucMap csucmap 38716   Suc csuccl 38717

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-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-cnv 5670  df-dm 5672  df-rn 5673  df-sucmap 39000  df-succl 39007

This theorem is referenced by:  dfsuccl3  39011