Theorem presucmap 38833

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 38834 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 38815	. . 3 ⊢ (𝑁 ∈ ran SucMap → pre 𝑁 = (℩𝑚𝑚 SucMap 𝑁))
2	1	eqcomd 2743	. 2 ⊢ (𝑁 ∈ ran SucMap → (℩𝑚𝑚 SucMap 𝑁) = pre 𝑁)
3		preex 38830	. . 3 ⊢ pre 𝑁 ∈ V
4		eupre2 38831	. . . 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 5089	. . . 4 ⊢ (𝑚 = pre 𝑁 → (𝑚 SucMap 𝑁 ↔ pre 𝑁 SucMap 𝑁))
7	6	iota2 6482	. . 3 ⊢ (( pre 𝑁 ∈ V ∧ ∃!𝑚 𝑚 SucMap 𝑁) → ( pre 𝑁 SucMap 𝑁 ↔ (℩𝑚𝑚 SucMap 𝑁) = pre 𝑁))
8	3, 5, 7	sylancr 588	. 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 1542   ∈ wcel 2114  ∃!weu 2569  Vcvv 3430   class class class wbr 5086  ran crn 5626  ℩cio 6447   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:  preuniqval  38834  sucpre  38835