Theorem dfsuccl2 38493

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 38492	. 2 ⊢ Suc = ran SucMap
2		df-sucmap 38485	. . 3 ⊢ SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
3	2	rneqi 5876	. 2 ⊢ ran SucMap = ran {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
4		rnopab 5893	. 2 ⊢ ran {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛} = {𝑛 ∣ ∃𝑚 suc 𝑚 = 𝑛}
5	1, 3, 4	3eqtri 2758	1 ⊢ Suc = {𝑛 ∣ ∃𝑚 suc 𝑚 = 𝑛}

Syntax hints:   = wceq 1541  ∃wex 1780  {cab 2709  {copab 5151  ran crn 5615  suc csuc 6308   SucMap csucmap 38227   Suc csuccl 38228

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-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-cnv 5622  df-dm 5624  df-rn 5625  df-sucmap 38485  df-succl 38492

This theorem is referenced by:  dfsuccl3  38496