Theorem dfpre4 39412

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 39394). (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 39407	. 2 ⊢ pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))
2		dfpred4 39411	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → Pred( SucMap , dom SucMap , 𝑁) = [𝑁]◡( SucMap ↾ dom SucMap ))
3		relsucmap 39399	. . . . . . . 8 ⊢ Rel SucMap
4		dfrel5 39278	. . . . . . . 8 ⊢ (Rel SucMap ↔ ( SucMap ↾ dom SucMap ) = SucMap )
5	3, 4	mpbi 233	. . . . . . 7 ⊢ ( SucMap ↾ dom SucMap ) = SucMap
6	5	cnveqi 5852	. . . . . 6 ⊢ ◡( SucMap ↾ dom SucMap ) = ◡ SucMap
7	6	eceq2i 8760	. . . . 5 ⊢ [𝑁]◡( SucMap ↾ dom SucMap ) = [𝑁]◡ SucMap
8	2, 7	eqtrdi 2812	. . . 4 ⊢ (𝑁 ∈ 𝑉 → Pred( SucMap , dom SucMap , 𝑁) = [𝑁]◡ SucMap )
9	8	eleq2d 2847	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁) ↔ 𝑚 ∈ [𝑁]◡ SucMap ))
10	9	iotabidv 6522	. 2 ⊢ (𝑁 ∈ 𝑉 → (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)) = (℩𝑚𝑚 ∈ [𝑁]◡ SucMap ))
11	1, 10	eqtrid 2808	1 ⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚𝑚 ∈ [𝑁]◡ SucMap ))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ^◡ccnv 5650  dom cdm 5651   ↾ cres 5653  Rel wrel 5656  Predcpred 6303  ℩cio 6492  [cec 8715   SucMap csucmap 39110   pre cpre 39112

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 2733  ax-sep 5249  ax-pr 5391

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 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  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 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-iota 6494  df-ec 8719  df-sucmap 39394  df-pre 39407

This theorem is used by: (None)