Theorem dfpre4 39678

Description: Alternate definition of the predecessor of the 𝑁 set. The ^◡ SucMap is just the "PreMap"; we did not define it because we do not expect to use it extensively in future (cf. the comments of df-sucmap 39660). (Contributed by Peter Mazsa, 26-Jan-2026.)

Assertion

Ref	Expression
dfpre4	⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚𝑚 ∈ [𝑁]◡ SucMap ))

Distinct variable groups:   𝑚,𝑁   𝑚,𝑉

Proof of Theorem dfpre4

Step	Hyp	Ref	Expression
1		df-pre 39673	. 2 ⊢ pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))
2		dfpred4 39677	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → Pred( SucMap , dom SucMap , 𝑁) = [𝑁]◡( SucMap ↾ dom SucMap ))
3		relsucmap 39665	. . . . . . . 8 ⊢ Rel SucMap
4		dfrel5 39544	. . . . . . . 8 ⊢ (Rel SucMap ↔ ( SucMap ↾ dom SucMap ) = SucMap )
5	3, 4	mpbi 233	. . . . . . 7 ⊢ ( SucMap ↾ dom SucMap ) = SucMap
6	5	cnveqi 5864	. . . . . 6 ⊢ ◡( SucMap ↾ dom SucMap ) = ◡ SucMap
7	6	eceq2i 8798	. . . . 5 ⊢ [𝑁]◡( SucMap ↾ dom SucMap ) = [𝑁]◡ SucMap
8	2, 7	eqtrdi 2814	. . . 4 ⊢ (𝑁 ∈ 𝑉 → Pred( SucMap , dom SucMap , 𝑁) = [𝑁]◡ SucMap )
9	8	eleq2d 2850	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁) ↔ 𝑚 ∈ [𝑁]◡ SucMap ))
10	9	iotabidv 6549	. 2 ⊢ (𝑁 ∈ 𝑉 → (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)) = (℩𝑚𝑚 ∈ [𝑁]◡ SucMap ))
11	1, 10	eqtrid 2810	1 ⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚𝑚 ∈ [𝑁]◡ SucMap ))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ^◡ccnv 5657  dom cdm 5658   ↾ cres 5660  Rel wrel 5663  Predcpred 6329  ℩cio 6518  [cec 8753   SucMap csucmap 39376   pre cpre 39378

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-cnv 5666  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-iota 6520  df-ec 8757  df-sucmap 39660  df-pre 39673

This theorem is used by: (None)