Theorem presucmap 39567

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 39568 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 39549	. . 3 ⊢ (𝑁 ∈ ran SucMap → pre 𝑁 = (℩𝑚𝑚 SucMap 𝑁))
2	1	eqcomd 2769	. 2 ⊢ (𝑁 ∈ ran SucMap → (℩𝑚𝑚 SucMap 𝑁) = pre 𝑁)
3		preex 39564	. . 3 ⊢ pre 𝑁 ∈ V
4		eupre2 39565	. . . 4 ⊢ (𝑁 ∈ ran SucMap → (𝑁 ∈ ran SucMap ↔ ∃!𝑚 𝑚 SucMap 𝑁))
5	4	ibi 270	. . 3 ⊢ (𝑁 ∈ ran SucMap → ∃!𝑚 𝑚 SucMap 𝑁)
6		breq1 5108	. . . 4 ⊢ (𝑚 = pre 𝑁 → (𝑚 SucMap 𝑁 ↔ pre 𝑁 SucMap 𝑁))
7	6	iota2 6534	. . 3 ⊢ (( pre 𝑁 ∈ V ∧ ∃!𝑚 𝑚 SucMap 𝑁) → ( pre 𝑁 SucMap 𝑁 ↔ (℩𝑚𝑚 SucMap 𝑁) = pre 𝑁))
8	3, 5, 7	sylancr 599	. 2 ⊢ (𝑁 ∈ ran SucMap → ( pre 𝑁 SucMap 𝑁 ↔ (℩𝑚𝑚 SucMap 𝑁) = pre 𝑁))
9	2, 8	mpbird 260	1 ⊢ (𝑁 ∈ ran SucMap → pre 𝑁 SucMap 𝑁)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2146  ∃!weu 2596  Vcvv 3453   class class class wbr 5105  ran crn 5655  ℩cio 6498   SucMap csucmap 39250   pre cpre 39252

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394  ax-un 7758  ax-reg 9596

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-eprel 5554  df-fr 5607  df-xp 5660  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-suc 6374  df-iota 6500  df-sucmap 39534  df-pre 39547

This theorem is used by:  preuniqval  39568  sucpre  39569