Theorem dfsuccl2 38837

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 38836	. 2 ⊢ Suc = ran SucMap
2		df-sucmap 38829	. . 3 ⊢ SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
3	2	rneqi 5879	. 2 ⊢ ran SucMap = ran {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
4		rnopab 5896	. 2 ⊢ ran {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛} = {𝑛 ∣ ∃𝑚 suc 𝑚 = 𝑛}
5	1, 3, 4	3eqtri 2766	1 ⊢ Suc = {𝑛 ∣ ∃𝑚 suc 𝑚 = 𝑛}

Syntax hints:   = wceq 1547  ∃wex 1786  {cab 2717  {copab 5134  ran crn 5619  suc csuc 6312   SucMap csucmap 38545   Suc csuccl 38546

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-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-cnv 5626  df-dm 5628  df-rn 5629  df-sucmap 38829  df-succl 38836

This theorem is referenced by:  dfsuccl3  38840