Theorem presucmap 38862

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 38863 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 38844	. . 3 ⊢ (𝑁 ∈ ran SucMap → pre 𝑁 = (℩𝑚𝑚 SucMap 𝑁))
2	1	eqcomd 2745	. 2 ⊢ (𝑁 ∈ ran SucMap → (℩𝑚𝑚 SucMap 𝑁) = pre 𝑁)
3		preex 38859	. . 3 ⊢ pre 𝑁 ∈ V
4		eupre2 38860	. . . 4 ⊢ (𝑁 ∈ ran SucMap → (𝑁 ∈ ran SucMap ↔ ∃!𝑚 𝑚 SucMap 𝑁))
5	4	ibi 268	. . 3 ⊢ (𝑁 ∈ ran SucMap → ∃!𝑚 𝑚 SucMap 𝑁)
6		breq1 5075	. . . 4 ⊢ (𝑚 = pre 𝑁 → (𝑚 SucMap 𝑁 ↔ pre 𝑁 SucMap 𝑁))
7	6	iota2 6474	. . 3 ⊢ (( pre 𝑁 ∈ V ∧ ∃!𝑚 𝑚 SucMap 𝑁) → ( pre 𝑁 SucMap 𝑁 ↔ (℩𝑚𝑚 SucMap 𝑁) = pre 𝑁))
8	3, 5, 7	sylancr 593	. 2 ⊢ (𝑁 ∈ ran SucMap → ( pre 𝑁 SucMap 𝑁 ↔ (℩𝑚𝑚 SucMap 𝑁) = pre 𝑁))
9	2, 8	mpbird 258	1 ⊢ (𝑁 ∈ ran SucMap → pre 𝑁 SucMap 𝑁)

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547   ∈ wcel 2119  ∃!weu 2572  Vcvv 3431   class class class wbr 5072  ran crn 5619  ℩cio 6439   SucMap csucmap 38545   pre cpre 38547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678  ax-reg 9497

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-eprel 5518  df-fr 5571  df-xp 5624  df-cnv 5626  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-suc 6316  df-iota 6441  df-sucmap 38829  df-pre 38842

This theorem is referenced by:  preuniqval  38863  sucpre  38864