Theorem presucmap 38517

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 38518 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 38500	. . 3 ⊢ (𝑁 ∈ ran SucMap → pre 𝑁 = (℩𝑚𝑚 SucMap 𝑁))
2	1	eqcomd 2737	. 2 ⊢ (𝑁 ∈ ran SucMap → (℩𝑚𝑚 SucMap 𝑁) = pre 𝑁)
3		preex 38514	. . 3 ⊢ pre 𝑁 ∈ V
4		eupre2 38515	. . . 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 5092	. . . 4 ⊢ (𝑚 = pre 𝑁 → (𝑚 SucMap 𝑁 ↔ pre 𝑁 SucMap 𝑁))
7	6	iota2 6470	. . 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 2111  ∃!weu 2563  Vcvv 3436   class class class wbr 5089  ran crn 5615  ℩cio 6435   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:  preuniqval  38518  sucpre  38519