Theorem dfsuccl2 38808

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 38807	. 2 ⊢ Suc = ran SucMap
2		df-sucmap 38800	. . 3 ⊢ SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
3	2	rneqi 5887	. 2 ⊢ ran SucMap = ran {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
4		rnopab 5904	. 2 ⊢ ran {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛} = {𝑛 ∣ ∃𝑚 suc 𝑚 = 𝑛}
5	1, 3, 4	3eqtri 2764	1 ⊢ Suc = {𝑛 ∣ ∃𝑚 suc 𝑚 = 𝑛}

Syntax hints:   = wceq 1542  ∃wex 1781  {cab 2715  {copab 5148  ran crn 5626  suc csuc 6320   SucMap csucmap 38516   Suc csuccl 38517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-cnv 5633  df-dm 5635  df-rn 5636  df-sucmap 38800  df-succl 38807

This theorem is referenced by:  dfsuccl3  38811