Theorem preuniqval 38834

Description: Uniqueness/canonicity of pre. presucmap 38833 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 38833	. . . 4 ⊢ (𝑁 ∈ ran SucMap → pre 𝑁 SucMap 𝑁)
2		preex 38830	. . . . . 6 ⊢ pre 𝑁 ∈ V
3		sucmapleftuniq 38828	. . . . . 6 ⊢ (( pre 𝑁 ∈ V ∧ 𝑚 ∈ V ∧ 𝑁 ∈ ran SucMap ) → (( pre 𝑁 SucMap 𝑁 ∧ 𝑚 SucMap 𝑁) → pre 𝑁 = 𝑚))
4	2, 3	mp3an1 1451	. . . . 5 ⊢ ((𝑚 ∈ V ∧ 𝑁 ∈ ran SucMap ) → (( pre 𝑁 SucMap 𝑁 ∧ 𝑚 SucMap 𝑁) → pre 𝑁 = 𝑚))
5	4	el2v1 38567	. . . 4 ⊢ (𝑁 ∈ ran SucMap → (( pre 𝑁 SucMap 𝑁 ∧ 𝑚 SucMap 𝑁) → pre 𝑁 = 𝑚))
6	1, 5	mpand 696	. . 3 ⊢ (𝑁 ∈ ran SucMap → (𝑚 SucMap 𝑁 → pre 𝑁 = 𝑚))
7		eqcom 2744	. . 3 ⊢ ( pre 𝑁 = 𝑚 ↔ 𝑚 = pre 𝑁)
8	6, 7	imbitrdi 251	. 2 ⊢ (𝑁 ∈ ran SucMap → (𝑚 SucMap 𝑁 → 𝑚 = pre 𝑁))
9	8	alrimiv 1929	1 ⊢ (𝑁 ∈ ran SucMap → ∀𝑚(𝑚 SucMap 𝑁 → 𝑚 = pre 𝑁))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  Vcvv 3430   class class class wbr 5086  ran crn 5626   SucMap csucmap 38516   pre cpre 38518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683  ax-reg 9501

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-eprel 5525  df-fr 5578  df-xp 5631  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-suc 6324  df-iota 6449  df-sucmap 38800  df-pre 38813

This theorem is referenced by: (None)