Theorem dfsucmap3 38830

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 2746	. . 3 ⊢ (𝑛 = suc 𝑚 ↔ suc 𝑚 = 𝑛)
2	1	opabbii 5139	. 2 ⊢ {⟨𝑚, 𝑛⟩ ∣ 𝑛 = suc 𝑚} = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
3		dfadjliftmap 38823	. . 3 ⊢ ( I AdjLiftMap V) = (𝑚 ∈ dom (( I ∪ ◡ E ) ↾ V) ↦ [𝑚](( I ∪ ◡ E ) ↾ V))
4		dmresv 6151	. . . . 5 ⊢ dom (( I ∪ ◡ E ) ↾ V) = dom ( I ∪ ◡ E )
5		dmun 5852	. . . . . 6 ⊢ dom ( I ∪ ◡ E ) = (dom I ∪ dom ◡ E )
6		dmi 5863	. . . . . . 7 ⊢ dom I = V
7		dmcnvep 38755	. . . . . . 7 ⊢ dom ◡ E = (V ∖ {∅})
8	6, 7	uneq12i 4096	. . . . . 6 ⊢ (dom I ∪ dom ◡ E ) = (V ∪ (V ∖ {∅}))
9		undifabs 4406	. . . . . 6 ⊢ (V ∪ (V ∖ {∅})) = V
10	5, 8, 9	3eqtri 2766	. . . . 5 ⊢ dom ( I ∪ ◡ E ) = V
11	4, 10	eqtri 2762	. . . 4 ⊢ dom (( I ∪ ◡ E ) ↾ V) = V
12		orcom 876	. . . . . . 7 ⊢ ((𝑛 ∈ {𝑚} ∨ 𝑛 ∈ 𝑚) ↔ (𝑛 ∈ 𝑚 ∨ 𝑛 ∈ {𝑚}))
13		elecALTV 38638	. . . . . . . . 9 ⊢ ((𝑚 ∈ V ∧ 𝑛 ∈ V) → (𝑛 ∈ [𝑚]( I ∪ ◡ E ) ↔ 𝑚( I ∪ ◡ E )𝑛))
14	13	el2v 3438	. . . . . . . 8 ⊢ (𝑛 ∈ [𝑚]( I ∪ ◡ E ) ↔ 𝑚( I ∪ ◡ E )𝑛)
15		brun 5123	. . . . . . . 8 ⊢ (𝑚( I ∪ ◡ E )𝑛 ↔ (𝑚 I 𝑛 ∨ 𝑚◡ E 𝑛))
16		equcom 2025	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 ↔ 𝑛 = 𝑚)
17		ideqg 5793	. . . . . . . . . . 11 ⊢ (𝑛 ∈ V → (𝑚 I 𝑛 ↔ 𝑚 = 𝑛))
18	17	elv 3436	. . . . . . . . . 10 ⊢ (𝑚 I 𝑛 ↔ 𝑚 = 𝑛)
19		velsn 4571	. . . . . . . . . 10 ⊢ (𝑛 ∈ {𝑚} ↔ 𝑛 = 𝑚)
20	16, 18, 19	3bitr4i 304	. . . . . . . . 9 ⊢ (𝑚 I 𝑛 ↔ 𝑛 ∈ {𝑚})
21		brcnvep 38637	. . . . . . . . . 10 ⊢ (𝑚 ∈ V → (𝑚◡ E 𝑛 ↔ 𝑛 ∈ 𝑚))
22	21	elv 3436	. . . . . . . . 9 ⊢ (𝑚◡ E 𝑛 ↔ 𝑛 ∈ 𝑚)
23	20, 22	orbi12i 920	. . . . . . . 8 ⊢ ((𝑚 I 𝑛 ∨ 𝑚◡ E 𝑛) ↔ (𝑛 ∈ {𝑚} ∨ 𝑛 ∈ 𝑚))
24	14, 15, 23	3bitri 298	. . . . . . 7 ⊢ (𝑛 ∈ [𝑚]( I ∪ ◡ E ) ↔ (𝑛 ∈ {𝑚} ∨ 𝑛 ∈ 𝑚))
25		elun 4083	. . . . . . 7 ⊢ (𝑛 ∈ (𝑚 ∪ {𝑚}) ↔ (𝑛 ∈ 𝑚 ∨ 𝑛 ∈ {𝑚}))
26	12, 24, 25	3bitr4i 304	. . . . . 6 ⊢ (𝑛 ∈ [𝑚]( I ∪ ◡ E ) ↔ 𝑛 ∈ (𝑚 ∪ {𝑚}))
27	26	eqriv 2736	. . . . 5 ⊢ [𝑚]( I ∪ ◡ E ) = (𝑚 ∪ {𝑚})
28		reli 5769	. . . . . . . 8 ⊢ Rel I
29		relcnv 6056	. . . . . . . 8 ⊢ Rel ◡ E
30		relun 5754	. . . . . . . 8 ⊢ (Rel ( I ∪ ◡ E ) ↔ (Rel I ∧ Rel ◡ E ))
31	28, 29, 30	mpbir2an 717	. . . . . . 7 ⊢ Rel ( I ∪ ◡ E )
32		dfrel3 6149	. . . . . . 7 ⊢ (Rel ( I ∪ ◡ E ) ↔ (( I ∪ ◡ E ) ↾ V) = ( I ∪ ◡ E ))
33	31, 32	mpbi 231	. . . . . 6 ⊢ (( I ∪ ◡ E ) ↾ V) = ( I ∪ ◡ E )
34	33	eceq2i 8676	. . . . 5 ⊢ [𝑚](( I ∪ ◡ E ) ↾ V) = [𝑚]( I ∪ ◡ E )
35		df-suc 6316	. . . . 5 ⊢ suc 𝑚 = (𝑚 ∪ {𝑚})
36	27, 34, 35	3eqtr4i 2772	. . . 4 ⊢ [𝑚](( I ∪ ◡ E ) ↾ V) = suc 𝑚
37	11, 36	mpteq12i 5169	. . 3 ⊢ (𝑚 ∈ dom (( I ∪ ◡ E ) ↾ V) ↦ [𝑚](( I ∪ ◡ E ) ↾ V)) = (𝑚 ∈ V ↦ suc 𝑚)
38		mptv 5178	. . 3 ⊢ (𝑚 ∈ V ↦ suc 𝑚) = {⟨𝑚, 𝑛⟩ ∣ 𝑛 = suc 𝑚}
39	3, 37, 38	3eqtri 2766	. 2 ⊢ ( I AdjLiftMap V) = {⟨𝑚, 𝑛⟩ ∣ 𝑛 = suc 𝑚}
40		df-sucmap 38829	. 2 ⊢ SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
41	2, 39, 40	3eqtr4ri 2773	1 ⊢ SucMap = ( I AdjLiftMap V)

Syntax hints:   ↔ wb 207   ∨ wo 853   = wceq 1547   ∈ wcel 2119  Vcvv 3431   ∖ cdif 3880   ∪ cun 3881  ∅c0 4261  {csn 4555   class class class wbr 5072  {copab 5134   ↦ cmpt 5153   I cid 5512   E cep 5517  ^◡ccnv 5617  dom cdm 5618   ↾ cres 5620  Rel wrel 5623  suc csuc 6312  [cec 8631   AdjLiftMap cadjliftmap 38543   SucMap csucmap 38545

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-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-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  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-mpt 5154  df-id 5513  df-eprel 5518  df-xp 5624  df-rel 5625  df-cnv 5626  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-suc 6316  df-ec 8635  df-qmap 38813  df-adjliftmap 38822  df-sucmap 38829

This theorem is referenced by:  dfsucmap2  38831