Theorem dfpre4 38731

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 38713). (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 38726	. 2 ⊢ pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))
2		dfpred4 38730	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → Pred( SucMap , dom SucMap , 𝑁) = [𝑁]◡( SucMap ↾ dom SucMap ))
3		relsucmap 38718	. . . . . . . 8 ⊢ Rel SucMap
4		dfrel5 38597	. . . . . . . 8 ⊢ (Rel SucMap ↔ ( SucMap ↾ dom SucMap ) = SucMap )
5	3, 4	mpbi 230	. . . . . . 7 ⊢ ( SucMap ↾ dom SucMap ) = SucMap
6	5	cnveqi 5831	. . . . . 6 ⊢ ◡( SucMap ↾ dom SucMap ) = ◡ SucMap
7	6	eceq2i 8688	. . . . 5 ⊢ [𝑁]◡( SucMap ↾ dom SucMap ) = [𝑁]◡ SucMap
8	2, 7	eqtrdi 2788	. . . 4 ⊢ (𝑁 ∈ 𝑉 → Pred( SucMap , dom SucMap , 𝑁) = [𝑁]◡ SucMap )
9	8	eleq2d 2823	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁) ↔ 𝑚 ∈ [𝑁]◡ SucMap ))
10	9	iotabidv 6484	. 2 ⊢ (𝑁 ∈ 𝑉 → (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)) = (℩𝑚𝑚 ∈ [𝑁]◡ SucMap ))
11	1, 10	eqtrid 2784	1 ⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚𝑚 ∈ [𝑁]◡ SucMap ))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ^◡ccnv 5631  dom cdm 5632   ↾ cres 5634  Rel wrel 5637  Predcpred 6266  ℩cio 6454  [cec 8643   SucMap csucmap 38429   pre cpre 38431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-iota 6456  df-ec 8647  df-sucmap 38713  df-pre 38726

This theorem is referenced by: (None)