Theorem dfpre4 39229

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 39211). (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 39224	. 2 ⊢ pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))
2		dfpred4 39228	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → Pred( SucMap , dom SucMap , 𝑁) = [𝑁]◡( SucMap ↾ dom SucMap ))
3		relsucmap 39216	. . . . . . . 8 ⊢ Rel SucMap
4		dfrel5 39095	. . . . . . . 8 ⊢ (Rel SucMap ↔ ( SucMap ↾ dom SucMap ) = SucMap )
5	3, 4	mpbi 233	. . . . . . 7 ⊢ ( SucMap ↾ dom SucMap ) = SucMap
6	5	cnveqi 5854	. . . . . 6 ⊢ ◡( SucMap ↾ dom SucMap ) = ◡ SucMap
7	6	eceq2i 8740	. . . . 5 ⊢ [𝑁]◡( SucMap ↾ dom SucMap ) = [𝑁]◡ SucMap
8	2, 7	eqtrdi 2811	. . . 4 ⊢ (𝑁 ∈ 𝑉 → Pred( SucMap , dom SucMap , 𝑁) = [𝑁]◡ SucMap )
9	8	eleq2d 2846	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁) ↔ 𝑚 ∈ [𝑁]◡ SucMap ))
10	9	iotabidv 6517	. 2 ⊢ (𝑁 ∈ 𝑉 → (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)) = (℩𝑚𝑚 ∈ [𝑁]◡ SucMap ))
11	1, 10	eqtrid 2807	1 ⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚𝑚 ∈ [𝑁]◡ SucMap ))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ^◡ccnv 5654  dom cdm 5655   ↾ cres 5657  Rel wrel 5660  Predcpred 6298  ℩cio 6487  [cec 8695   SucMap csucmap 38927   pre cpre 38929

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  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 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-iota 6489  df-ec 8699  df-sucmap 39211  df-pre 39224

This theorem is used by: (None)