Theorem preuniqval 38747

Description: Uniqueness/canonicity of pre. presucmap 38746 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 38746	. . . 4 ⊢ (𝑁 ∈ ran SucMap → pre 𝑁 SucMap 𝑁)
2		preex 38743	. . . . . 6 ⊢ pre 𝑁 ∈ V
3		sucmapleftuniq 38741	. . . . . 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 38480	. . . 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 3442   class class class wbr 5100  ran crn 5633   SucMap csucmap 38429   pre cpre 38431

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 5243  ax-nul 5253  ax-pr 5379  ax-un 7690  ax-reg 9509

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 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-eprel 5532  df-fr 5585  df-xp 5638  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-suc 6331  df-iota 6456  df-sucmap 38713  df-pre 38726

This theorem is referenced by: (None)