Theorem preuniqval 38518

Description: Uniqueness/canonicity of pre. presucmap 38517 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 38517	. . . 4 ⊢ (𝑁 ∈ ran SucMap → pre 𝑁 SucMap 𝑁)
2		preex 38514	. . . . . 6 ⊢ pre 𝑁 ∈ V
3		sucmapleftuniq 38512	. . . . . 6 ⊢ (( pre 𝑁 ∈ V ∧ 𝑚 ∈ V ∧ 𝑁 ∈ ran SucMap ) → (( pre 𝑁 SucMap 𝑁 ∧ 𝑚 SucMap 𝑁) → pre 𝑁 = 𝑚))
4	2, 3	mp3an1 1450	. . . . 5 ⊢ ((𝑚 ∈ V ∧ 𝑁 ∈ ran SucMap ) → (( pre 𝑁 SucMap 𝑁 ∧ 𝑚 SucMap 𝑁) → pre 𝑁 = 𝑚))
5	4	el2v1 38274	. . . 4 ⊢ (𝑁 ∈ ran SucMap → (( pre 𝑁 SucMap 𝑁 ∧ 𝑚 SucMap 𝑁) → pre 𝑁 = 𝑚))
6	1, 5	mpand 695	. . 3 ⊢ (𝑁 ∈ ran SucMap → (𝑚 SucMap 𝑁 → pre 𝑁 = 𝑚))
7		eqcom 2738	. . 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 1928	1 ⊢ (𝑁 ∈ ran SucMap → ∀𝑚(𝑚 SucMap 𝑁 → 𝑚 = pre 𝑁))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2111  Vcvv 3436   class class class wbr 5089  ran crn 5615   SucMap csucmap 38227   pre cpre 38229

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668  ax-reg 9478

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-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-eprel 5514  df-fr 5567  df-xp 5620  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-suc 6312  df-iota 6437  df-sucmap 38485  df-pre 38498

This theorem is referenced by: (None)