Theorem dfsucmap3 39219

Description: Alternate definition of the successor map. (Contributed by Peter Mazsa, 28-Jan-2026.)

Assertion

Ref	Expression
dfsucmap3	⊢ SucMap = ( I AdjLiftMap V)

Proof of Theorem dfsucmap3

Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqcom 2769	. . 3 ⊢ (𝑛 = suc 𝑚 ↔ suc 𝑚 = 𝑛)
2	1	opabbii 5176	. 2 ⊢ {⟨𝑚, 𝑛⟩ ∣ 𝑛 = suc 𝑚} = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
3		dfadjliftmap 39212	. . 3 ⊢ ( I AdjLiftMap V) = (𝑚 ∈ dom (( I ∪ ◡ E ) ↾ V) ↦ [𝑚](( I ∪ ◡ E ) ↾ V))
4		dmresv 6198	. . . . 5 ⊢ dom (( I ∪ ◡ E ) ↾ V) = dom ( I ∪ ◡ E )
5		dmun 5898	. . . . . 6 ⊢ dom ( I ∪ ◡ E ) = (dom I ∪ dom ◡ E )
6		dmi 5909	. . . . . . 7 ⊢ dom I = V
7		dmcnvep 39144	. . . . . . 7 ⊢ dom ◡ E = (V ∖ {∅})
8	6, 7	uneq12i 4116	. . . . . 6 ⊢ (dom I ∪ dom ◡ E ) = (V ∪ (V ∖ {∅}))
9		undifabs 4437	. . . . . 6 ⊢ (V ∪ (V ∖ {∅})) = V
10	5, 8, 9	3eqtri 2789	. . . . 5 ⊢ dom ( I ∪ ◡ E ) = V
11	4, 10	eqtri 2785	. . . 4 ⊢ dom (( I ∪ ◡ E ) ↾ V) = V
12		orcom 884	. . . . . . 7 ⊢ ((𝑛 ∈ {𝑚} ∨ 𝑛 ∈ 𝑚) ↔ (𝑛 ∈ 𝑚 ∨ 𝑛 ∈ {𝑚}))
13		elecALTV 39027	. . . . . . . . 9 ⊢ ((𝑚 ∈ V ∧ 𝑛 ∈ V) → (𝑛 ∈ [𝑚]( I ∪ ◡ E ) ↔ 𝑚( I ∪ ◡ E )𝑛))
14	13	el2v 3460	. . . . . . . 8 ⊢ (𝑛 ∈ [𝑚]( I ∪ ◡ E ) ↔ 𝑚( I ∪ ◡ E )𝑛)
15		brun 5160	. . . . . . . 8 ⊢ (𝑚( I ∪ ◡ E )𝑛 ↔ (𝑚 I 𝑛 ∨ 𝑚◡ E 𝑛))
16		equcom 2051	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 ↔ 𝑛 = 𝑚)
17		ideqg 5835	. . . . . . . . . . 11 ⊢ (𝑛 ∈ V → (𝑚 I 𝑛 ↔ 𝑚 = 𝑛))
18	17	elv 3458	. . . . . . . . . 10 ⊢ (𝑚 I 𝑛 ↔ 𝑚 = 𝑛)
19		velsn 4603	. . . . . . . . . 10 ⊢ (𝑛 ∈ {𝑚} ↔ 𝑛 = 𝑚)
20	16, 18, 19	3bitr4i 306	. . . . . . . . 9 ⊢ (𝑚 I 𝑛 ↔ 𝑛 ∈ {𝑚})
21		brcnvep 39026	. . . . . . . . . 10 ⊢ (𝑚 ∈ V → (𝑚◡ E 𝑛 ↔ 𝑛 ∈ 𝑚))
22	21	elv 3458	. . . . . . . . 9 ⊢ (𝑚◡ E 𝑛 ↔ 𝑛 ∈ 𝑚)
23	20, 22	orbi12i 928	. . . . . . . 8 ⊢ ((𝑚 I 𝑛 ∨ 𝑚◡ E 𝑛) ↔ (𝑛 ∈ {𝑚} ∨ 𝑛 ∈ 𝑚))
24	14, 15, 23	3bitri 300	. . . . . . 7 ⊢ (𝑛 ∈ [𝑚]( I ∪ ◡ E ) ↔ (𝑛 ∈ {𝑚} ∨ 𝑛 ∈ 𝑚))
25		elun 4103	. . . . . . 7 ⊢ (𝑛 ∈ (𝑚 ∪ {𝑚}) ↔ (𝑛 ∈ 𝑚 ∨ 𝑛 ∈ {𝑚}))
26	12, 24, 25	3bitr4i 306	. . . . . 6 ⊢ (𝑛 ∈ [𝑚]( I ∪ ◡ E ) ↔ 𝑛 ∈ (𝑚 ∪ {𝑚}))
27	26	eqriv 2759	. . . . 5 ⊢ [𝑚]( I ∪ ◡ E ) = (𝑚 ∪ {𝑚})
28		reli 5811	. . . . . . . 8 ⊢ Rel I
29		relcnv 6104	. . . . . . . 8 ⊢ Rel ◡ E
30		relun 5796	. . . . . . . 8 ⊢ (Rel ( I ∪ ◡ E ) ↔ (Rel I ∧ Rel ◡ E ))
31	28, 29, 30	mpbir2an 724	. . . . . . 7 ⊢ Rel ( I ∪ ◡ E )
32		dfrel3 6196	. . . . . . 7 ⊢ (Rel ( I ∪ ◡ E ) ↔ (( I ∪ ◡ E ) ↾ V) = ( I ∪ ◡ E ))
33	31, 32	mpbi 233	. . . . . 6 ⊢ (( I ∪ ◡ E ) ↾ V) = ( I ∪ ◡ E )
34	33	eceq2i 8743	. . . . 5 ⊢ [𝑚](( I ∪ ◡ E ) ↾ V) = [𝑚]( I ∪ ◡ E )
35		df-suc 6367	. . . . 5 ⊢ suc 𝑚 = (𝑚 ∪ {𝑚})
36	27, 34, 35	3eqtr4i 2795	. . . 4 ⊢ [𝑚](( I ∪ ◡ E ) ↾ V) = suc 𝑚
37	11, 36	mpteq12i 5206	. . 3 ⊢ (𝑚 ∈ dom (( I ∪ ◡ E ) ↾ V) ↦ [𝑚](( I ∪ ◡ E ) ↾ V)) = (𝑚 ∈ V ↦ suc 𝑚)
38		mptv 5215	. . 3 ⊢ (𝑚 ∈ V ↦ suc 𝑚) = {⟨𝑚, 𝑛⟩ ∣ 𝑛 = suc 𝑚}
39	3, 37, 38	3eqtri 2789	. 2 ⊢ ( I AdjLiftMap V) = {⟨𝑚, 𝑛⟩ ∣ 𝑛 = suc 𝑚}
40		df-sucmap 39218	. 2 ⊢ SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
41	2, 39, 40	3eqtr4ri 2796	1 ⊢ SucMap = ( I AdjLiftMap V)

This proof depends on syntax axioms:   ↔ wb 209   ∨ wo 861   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ∖ cdif 3899   ∪ cun 3900  ∅c0 4282  {csn 4587   class class class wbr 5107  {copab 5171   ↦ cmpt 5190   I cid 5553   E cep 5558  ^◡ccnv 5658  dom cdm 5659   ↾ cres 5661  Rel wrel 5664  suc csuc 6363  [cec 8698   AdjLiftMap cadjliftmap 38932   SucMap csucmap 38934

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-10 2178  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-eprel 5559  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-suc 6367  df-ec 8702  df-qmap 39202  df-adjliftmap 39211  df-sucmap 39218

This theorem is used by:  dfsucmap2  39220