Theorem dfpre4 38503

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 38485). (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 38498	. 2 ⊢ pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))
2		dfpred4 38502	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → Pred( SucMap , dom SucMap , 𝑁) = [𝑁]◡( SucMap ↾ dom SucMap ))
3		relsucmap 38490	. . . . . . . 8 ⊢ Rel SucMap
4		dfrel5 38388	. . . . . . . 8 ⊢ (Rel SucMap ↔ ( SucMap ↾ dom SucMap ) = SucMap )
5	3, 4	mpbi 230	. . . . . . 7 ⊢ ( SucMap ↾ dom SucMap ) = SucMap
6	5	cnveqi 5813	. . . . . 6 ⊢ ◡( SucMap ↾ dom SucMap ) = ◡ SucMap
7	6	eceq2i 8664	. . . . 5 ⊢ [𝑁]◡( SucMap ↾ dom SucMap ) = [𝑁]◡ SucMap
8	2, 7	eqtrdi 2782	. . . 4 ⊢ (𝑁 ∈ 𝑉 → Pred( SucMap , dom SucMap , 𝑁) = [𝑁]◡ SucMap )
9	8	eleq2d 2817	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁) ↔ 𝑚 ∈ [𝑁]◡ SucMap ))
10	9	iotabidv 6465	. 2 ⊢ (𝑁 ∈ 𝑉 → (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)) = (℩𝑚𝑚 ∈ [𝑁]◡ SucMap ))
11	1, 10	eqtrid 2778	1 ⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚𝑚 ∈ [𝑁]◡ SucMap ))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ^◡ccnv 5613  dom cdm 5614   ↾ cres 5616  Rel wrel 5619  Predcpred 6247  ℩cio 6435  [cec 8620   SucMap csucmap 38227   pre cpre 38229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-iota 6437  df-ec 8624  df-sucmap 38485  df-pre 38498

This theorem is referenced by: (None)