Theorem dfsucmap3 38576

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 2741	. . 3 ⊢ (𝑛 = suc 𝑚 ↔ suc 𝑚 = 𝑛)
2	1	opabbii 5163	. 2 ⊢ {⟨𝑚, 𝑛⟩ ∣ 𝑛 = suc 𝑚} = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
3		df-adjliftmap 38570	. . 3 ⊢ ( I AdjLiftMap V) = (𝑚 ∈ dom (( I ∪ ◡ E ) ↾ V) ↦ [𝑚](( I ∪ ◡ E ) ↾ V))
4		dmresv 6156	. . . . 5 ⊢ dom (( I ∪ ◡ E ) ↾ V) = dom ( I ∪ ◡ E )
5		dmun 5857	. . . . . 6 ⊢ dom ( I ∪ ◡ E ) = (dom I ∪ dom ◡ E )
6		dmi 5868	. . . . . . 7 ⊢ dom I = V
7		dmcnvep 38512	. . . . . . 7 ⊢ dom ◡ E = (V ∖ {∅})
8	6, 7	uneq12i 4116	. . . . . 6 ⊢ (dom I ∪ dom ◡ E ) = (V ∪ (V ∖ {∅}))
9		undifabs 4428	. . . . . 6 ⊢ (V ∪ (V ∖ {∅})) = V
10	5, 8, 9	3eqtri 2761	. . . . 5 ⊢ dom ( I ∪ ◡ E ) = V
11	4, 10	eqtri 2757	. . . 4 ⊢ dom (( I ∪ ◡ E ) ↾ V) = V
12		orcom 870	. . . . . . 7 ⊢ ((𝑛 ∈ {𝑚} ∨ 𝑛 ∈ 𝑚) ↔ (𝑛 ∈ 𝑚 ∨ 𝑛 ∈ {𝑚}))
13		elecALTV 38403	. . . . . . . . 9 ⊢ ((𝑚 ∈ V ∧ 𝑛 ∈ V) → (𝑛 ∈ [𝑚]( I ∪ ◡ E ) ↔ 𝑚( I ∪ ◡ E )𝑛))
14	13	el2v 3445	. . . . . . . 8 ⊢ (𝑛 ∈ [𝑚]( I ∪ ◡ E ) ↔ 𝑚( I ∪ ◡ E )𝑛)
15		brun 5147	. . . . . . . 8 ⊢ (𝑚( I ∪ ◡ E )𝑛 ↔ (𝑚 I 𝑛 ∨ 𝑚◡ E 𝑛))
16		equcom 2019	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 ↔ 𝑛 = 𝑚)
17		ideqg 5798	. . . . . . . . . . 11 ⊢ (𝑛 ∈ V → (𝑚 I 𝑛 ↔ 𝑚 = 𝑛))
18	17	elv 3443	. . . . . . . . . 10 ⊢ (𝑚 I 𝑛 ↔ 𝑚 = 𝑛)
19		velsn 4594	. . . . . . . . . 10 ⊢ (𝑛 ∈ {𝑚} ↔ 𝑛 = 𝑚)
20	16, 18, 19	3bitr4i 303	. . . . . . . . 9 ⊢ (𝑚 I 𝑛 ↔ 𝑛 ∈ {𝑚})
21		brcnvep 38402	. . . . . . . . . 10 ⊢ (𝑚 ∈ V → (𝑚◡ E 𝑛 ↔ 𝑛 ∈ 𝑚))
22	21	elv 3443	. . . . . . . . 9 ⊢ (𝑚◡ E 𝑛 ↔ 𝑛 ∈ 𝑚)
23	20, 22	orbi12i 914	. . . . . . . 8 ⊢ ((𝑚 I 𝑛 ∨ 𝑚◡ E 𝑛) ↔ (𝑛 ∈ {𝑚} ∨ 𝑛 ∈ 𝑚))
24	14, 15, 23	3bitri 297	. . . . . . 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 303	. . . . . 6 ⊢ (𝑛 ∈ [𝑚]( I ∪ ◡ E ) ↔ 𝑛 ∈ (𝑚 ∪ {𝑚}))
27	26	eqriv 2731	. . . . 5 ⊢ [𝑚]( I ∪ ◡ E ) = (𝑚 ∪ {𝑚})
28		reli 5773	. . . . . . . 8 ⊢ Rel I
29		relcnv 6061	. . . . . . . 8 ⊢ Rel ◡ E
30		relun 5758	. . . . . . . 8 ⊢ (Rel ( I ∪ ◡ E ) ↔ (Rel I ∧ Rel ◡ E ))
31	28, 29, 30	mpbir2an 711	. . . . . . 7 ⊢ Rel ( I ∪ ◡ E )
32		dfrel3 6154	. . . . . . 7 ⊢ (Rel ( I ∪ ◡ E ) ↔ (( I ∪ ◡ E ) ↾ V) = ( I ∪ ◡ E ))
33	31, 32	mpbi 230	. . . . . 6 ⊢ (( I ∪ ◡ E ) ↾ V) = ( I ∪ ◡ E )
34	33	eceq2i 8675	. . . . 5 ⊢ [𝑚](( I ∪ ◡ E ) ↾ V) = [𝑚]( I ∪ ◡ E )
35		df-suc 6321	. . . . 5 ⊢ suc 𝑚 = (𝑚 ∪ {𝑚})
36	27, 34, 35	3eqtr4i 2767	. . . 4 ⊢ [𝑚](( I ∪ ◡ E ) ↾ V) = suc 𝑚
37	11, 36	mpteq12i 5193	. . 3 ⊢ (𝑚 ∈ dom (( I ∪ ◡ E ) ↾ V) ↦ [𝑚](( I ∪ ◡ E ) ↾ V)) = (𝑚 ∈ V ↦ suc 𝑚)
38		mptv 5202	. . 3 ⊢ (𝑚 ∈ V ↦ suc 𝑚) = {⟨𝑚, 𝑛⟩ ∣ 𝑛 = suc 𝑚}
39	3, 37, 38	3eqtri 2761	. 2 ⊢ ( I AdjLiftMap V) = {⟨𝑚, 𝑛⟩ ∣ 𝑛 = suc 𝑚}
40		df-sucmap 38575	. 2 ⊢ SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
41	2, 39, 40	3eqtr4ri 2768	1 ⊢ SucMap = ( I AdjLiftMap V)

Syntax hints:   ↔ wb 206   ∨ wo 847   = wceq 1541   ∈ wcel 2113  Vcvv 3438   ∖ cdif 3896   ∪ cun 3897  ∅c0 4283  {csn 4578   class class class wbr 5096  {copab 5158   ↦ cmpt 5177   I cid 5516   E cep 5521  ^◡ccnv 5621  dom cdm 5622   ↾ cres 5624  Rel wrel 5627  suc csuc 6317  [cec 8631   AdjLiftMap cadjliftmap 38315   SucMap csucmap 38317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-eprel 5522  df-xp 5628  df-rel 5629  df-cnv 5630  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-suc 6321  df-ec 8635  df-adjliftmap 38570  df-sucmap 38575

This theorem is referenced by:  dfsucmap2  38577