Theorem preuniqval 38863

Description: Uniqueness/canonicity of pre. presucmap 38862 gives one witness; this theorem gives it is the only one. It turns any predecessor proof into an equality with pre 𝑁. (Contributed by Peter Mazsa, 12-Jan-2026.)

Assertion

Ref	Expression
preuniqval	⊢ (𝑁 ∈ ran SucMap → ∀𝑚(𝑚 SucMap 𝑁 → 𝑚 = pre 𝑁))

Distinct variable group:   𝑚,𝑁

Proof of Theorem preuniqval

Step	Hyp	Ref	Expression
1		presucmap 38862	. . . 4 ⊢ (𝑁 ∈ ran SucMap → pre 𝑁 SucMap 𝑁)
2		preex 38859	. . . . . 6 ⊢ pre 𝑁 ∈ V
3		sucmapleftuniq 38857	. . . . . 6 ⊢ (( pre 𝑁 ∈ V ∧ 𝑚 ∈ V ∧ 𝑁 ∈ ran SucMap ) → (( pre 𝑁 SucMap 𝑁 ∧ 𝑚 SucMap 𝑁) → pre 𝑁 = 𝑚))
4	2, 3	mp3an1 1456	. . . . 5 ⊢ ((𝑚 ∈ V ∧ 𝑁 ∈ ran SucMap ) → (( pre 𝑁 SucMap 𝑁 ∧ 𝑚 SucMap 𝑁) → pre 𝑁 = 𝑚))
5	4	el2v1 38596	. . . 4 ⊢ (𝑁 ∈ ran SucMap → (( pre 𝑁 SucMap 𝑁 ∧ 𝑚 SucMap 𝑁) → pre 𝑁 = 𝑚))
6	1, 5	mpand 701	. . 3 ⊢ (𝑁 ∈ ran SucMap → (𝑚 SucMap 𝑁 → pre 𝑁 = 𝑚))
7		eqcom 2746	. . 3 ⊢ ( pre 𝑁 = 𝑚 ↔ 𝑚 = pre 𝑁)
8	6, 7	imbitrdi 252	. 2 ⊢ (𝑁 ∈ ran SucMap → (𝑚 SucMap 𝑁 → 𝑚 = pre 𝑁))
9	8	alrimiv 1934	1 ⊢ (𝑁 ∈ ran SucMap → ∀𝑚(𝑚 SucMap 𝑁 → 𝑚 = pre 𝑁))

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1545   = wceq 1547   ∈ wcel 2119  Vcvv 3431   class class class wbr 5072  ran crn 5619   SucMap csucmap 38545   pre cpre 38547

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-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678  ax-reg 9497

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-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  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-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-eprel 5518  df-fr 5571  df-xp 5624  df-cnv 5626  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-suc 6316  df-iota 6441  df-sucmap 38829  df-pre 38842

This theorem is referenced by: (None)