Theorem presucmap 38668

Description: pre is really a predecessor (when it should be). This correctness theorem for pre makes it usable in proofs without unfolding ℩. This theorem gives one witness; preuniqval 38669 gives it is the only one. (Contributed by Peter Mazsa, 12-Jan-2026.)

Assertion

Ref	Expression
presucmap	⊢ (𝑁 ∈ ran SucMap → pre 𝑁 SucMap 𝑁)

Proof of Theorem presucmap

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfpre2 38651	. . 3 ⊢ (𝑁 ∈ ran SucMap → pre 𝑁 = (℩𝑚𝑚 SucMap 𝑁))
2	1	eqcomd 2742	. 2 ⊢ (𝑁 ∈ ran SucMap → (℩𝑚𝑚 SucMap 𝑁) = pre 𝑁)
3		preex 38665	. . 3 ⊢ pre 𝑁 ∈ V
4		eupre2 38666	. . . 4 ⊢ (𝑁 ∈ ran SucMap → (𝑁 ∈ ran SucMap ↔ ∃!𝑚 𝑚 SucMap 𝑁))
5	4	ibi 267	. . 3 ⊢ (𝑁 ∈ ran SucMap → ∃!𝑚 𝑚 SucMap 𝑁)
6		breq1 5101	. . . 4 ⊢ (𝑚 = pre 𝑁 → (𝑚 SucMap 𝑁 ↔ pre 𝑁 SucMap 𝑁))
7	6	iota2 6481	. . 3 ⊢ (( pre 𝑁 ∈ V ∧ ∃!𝑚 𝑚 SucMap 𝑁) → ( pre 𝑁 SucMap 𝑁 ↔ (℩𝑚𝑚 SucMap 𝑁) = pre 𝑁))
8	3, 5, 7	sylancr 587	. 2 ⊢ (𝑁 ∈ ran SucMap → ( pre 𝑁 SucMap 𝑁 ↔ (℩𝑚𝑚 SucMap 𝑁) = pre 𝑁))
9	2, 8	mpbird 257	1 ⊢ (𝑁 ∈ ran SucMap → pre 𝑁 SucMap 𝑁)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113  ∃!weu 2568  Vcvv 3440   class class class wbr 5098  ran crn 5625  ℩cio 6446   SucMap csucmap 38378   pre cpre 38380

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-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680  ax-reg 9497

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-eprel 5524  df-fr 5577  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-suc 6323  df-iota 6448  df-sucmap 38636  df-pre 38649

This theorem is referenced by:  preuniqval  38669  sucpre  38670