Theorem dfsucmap3 39112

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 2770	. . 3 ⊢ (𝑛 = suc 𝑚 ↔ suc 𝑚 = 𝑛)
2	1	opabbii 5178	. 2 ⊢ {⟨𝑚, 𝑛⟩ ∣ 𝑛 = suc 𝑚} = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
3		dfadjliftmap 39105	. . 3 ⊢ ( I AdjLiftMap V) = (𝑚 ∈ dom (( I ∪ ◡ E ) ↾ V) ↦ [𝑚](( I ∪ ◡ E ) ↾ V))
4		dmresv 6199	. . . . 5 ⊢ dom (( I ∪ ◡ E ) ↾ V) = dom ( I ∪ ◡ E )
5		dmun 5900	. . . . . 6 ⊢ dom ( I ∪ ◡ E ) = (dom I ∪ dom ◡ E )
6		dmi 5911	. . . . . . 7 ⊢ dom I = V
7		dmcnvep 39037	. . . . . . 7 ⊢ dom ◡ E = (V ∖ {∅})
8	6, 7	uneq12i 4120	. . . . . 6 ⊢ (dom I ∪ dom ◡ E ) = (V ∪ (V ∖ {∅}))
9		undifabs 4439	. . . . . 6 ⊢ (V ∪ (V ∖ {∅})) = V
10	5, 8, 9	3eqtri 2790	. . . . 5 ⊢ dom ( I ∪ ◡ E ) = V
11	4, 10	eqtri 2786	. . . 4 ⊢ dom (( I ∪ ◡ E ) ↾ V) = V
12		orcom 883	. . . . . . 7 ⊢ ((𝑛 ∈ {𝑚} ∨ 𝑛 ∈ 𝑚) ↔ (𝑛 ∈ 𝑚 ∨ 𝑛 ∈ {𝑚}))
13		elecALTV 38920	. . . . . . . . 9 ⊢ ((𝑚 ∈ V ∧ 𝑛 ∈ V) → (𝑛 ∈ [𝑚]( I ∪ ◡ E ) ↔ 𝑚( I ∪ ◡ E )𝑛))
14	13	el2v 3462	. . . . . . . 8 ⊢ (𝑛 ∈ [𝑚]( I ∪ ◡ E ) ↔ 𝑚( I ∪ ◡ E )𝑛)
15		brun 5162	. . . . . . . 8 ⊢ (𝑚( I ∪ ◡ E )𝑛 ↔ (𝑚 I 𝑛 ∨ 𝑚◡ E 𝑛))
16		equcom 2048	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 ↔ 𝑛 = 𝑚)
17		ideqg 5837	. . . . . . . . . . 11 ⊢ (𝑛 ∈ V → (𝑚 I 𝑛 ↔ 𝑚 = 𝑛))
18	17	elv 3460	. . . . . . . . . 10 ⊢ (𝑚 I 𝑛 ↔ 𝑚 = 𝑛)
19		velsn 4605	. . . . . . . . . 10 ⊢ (𝑛 ∈ {𝑚} ↔ 𝑛 = 𝑚)
20	16, 18, 19	3bitr4i 306	. . . . . . . . 9 ⊢ (𝑚 I 𝑛 ↔ 𝑛 ∈ {𝑚})
21		brcnvep 38919	. . . . . . . . . 10 ⊢ (𝑚 ∈ V → (𝑚◡ E 𝑛 ↔ 𝑛 ∈ 𝑚))
22	21	elv 3460	. . . . . . . . 9 ⊢ (𝑚◡ E 𝑛 ↔ 𝑛 ∈ 𝑚)
23	20, 22	orbi12i 927	. . . . . . . 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 4107	. . . . . . 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 2760	. . . . 5 ⊢ [𝑚]( I ∪ ◡ E ) = (𝑚 ∪ {𝑚})
28		reli 5813	. . . . . . . 8 ⊢ Rel I
29		relcnv 6106	. . . . . . . 8 ⊢ Rel ◡ E
30		relun 5798	. . . . . . . 8 ⊢ (Rel ( I ∪ ◡ E ) ↔ (Rel I ∧ Rel ◡ E ))
31	28, 29, 30	mpbir2an 723	. . . . . . 7 ⊢ Rel ( I ∪ ◡ E )
32		dfrel3 6197	. . . . . . 7 ⊢ (Rel ( I ∪ ◡ E ) ↔ (( I ∪ ◡ E ) ↾ V) = ( I ∪ ◡ E ))
33	31, 32	mpbi 233	. . . . . 6 ⊢ (( I ∪ ◡ E ) ↾ V) = ( I ∪ ◡ E )
34	33	eceq2i 8733	. . . . 5 ⊢ [𝑚](( I ∪ ◡ E ) ↾ V) = [𝑚]( I ∪ ◡ E )
35		df-suc 6366	. . . . 5 ⊢ suc 𝑚 = (𝑚 ∪ {𝑚})
36	27, 34, 35	3eqtr4i 2796	. . . 4 ⊢ [𝑚](( I ∪ ◡ E ) ↾ V) = suc 𝑚
37	11, 36	mpteq12i 5208	. . 3 ⊢ (𝑚 ∈ dom (( I ∪ ◡ E ) ↾ V) ↦ [𝑚](( I ∪ ◡ E ) ↾ V)) = (𝑚 ∈ V ↦ suc 𝑚)
38		mptv 5217	. . 3 ⊢ (𝑚 ∈ V ↦ suc 𝑚) = {⟨𝑚, 𝑛⟩ ∣ 𝑛 = suc 𝑚}
39	3, 37, 38	3eqtri 2790	. 2 ⊢ ( I AdjLiftMap V) = {⟨𝑚, 𝑛⟩ ∣ 𝑛 = suc 𝑚}
40		df-sucmap 39111	. 2 ⊢ SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
41	2, 39, 40	3eqtr4ri 2797	1 ⊢ SucMap = ( I AdjLiftMap V)

Syntax hints:   ↔ wb 209   ∨ wo 860   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∖ cdif 3902   ∪ cun 3903  ∅c0 4286  {csn 4589   class class class wbr 5109  {copab 5173   ↦ cmpt 5192   I cid 5555   E cep 5560  ^◡ccnv 5660  dom cdm 5661   ↾ cres 5663  Rel wrel 5666  suc csuc 6362  [cec 8688   AdjLiftMap cadjliftmap 38825   SucMap csucmap 38827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-suc 6366  df-ec 8692  df-qmap 39095  df-adjliftmap 39104  df-sucmap 39111

This theorem is referenced by:  dfsucmap2  39113