dfsucmap2 - Mathbox for Peter Mazsa

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Theorem dfsucmap2 38831

Description: Alternate definition of the successor map. (Contributed by Peter Mazsa, 28-Jan-2026.)

**Assertion**
Ref	Expression
dfsucmap2	⊢ SucMap = ( I AdjLiftMap dom I )

**Proof of Theorem dfsucmap2**
Step	Hyp	Ref	Expression
1		dfsucmap3 38830	. 2 ⊢ SucMap = ( I AdjLiftMap V)
2		dmi 5863	. . . . . 6 ⊢ dom I = V
3	2	reseq2i 5928	. . . . 5 ⊢ (( I ∪ ^◡ E ) ↾ dom I ) = (( I ∪ ^◡ E ) ↾ V)
4	3	dmeqi 5846	. . . 4 ⊢ dom (( I ∪ ^◡ E ) ↾ dom I ) = dom (( I ∪ ^◡ E ) ↾ V)
5	3	eceq2i 8676	. . . 4 ⊢ [𝑚](( I ∪ ^◡ E ) ↾ dom I ) = [𝑚](( I ∪ ^◡ E ) ↾ V)
6	4, 5	mpteq12i 5169	. . 3 ⊢ (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ dom I ) ↦ [𝑚](( I ∪ ^◡ E ) ↾ dom I )) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ V) ↦ [𝑚](( I ∪ ^◡ E ) ↾ V))
7		dfadjliftmap 38823	. . 3 ⊢ ( I AdjLiftMap dom I ) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ dom I ) ↦ [𝑚](( I ∪ ^◡ E ) ↾ dom I ))
8		dfadjliftmap 38823	. . 3 ⊢ ( I AdjLiftMap V) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ V) ↦ [𝑚](( I ∪ ^◡ E ) ↾ V))
9	6, 7, 8	3eqtr4i 2772	. 2 ⊢ ( I AdjLiftMap dom I ) = ( I AdjLiftMap V)
10	1, 9	eqtr4i 2765	1 ⊢ SucMap = ( I AdjLiftMap dom I )

Colors of variables: wff setvar class

Syntax hints: = wceq 1547 Vcvv 3431 ∪ cun 3881 ↦ cmpt 5153 I cid 5512 E cep 5517 ^◡ccnv 5617 dom cdm 5618 ↾ cres 5620 [cec 8631 AdjLiftMap cadjliftmap 38543 SucMap csucmap 38545

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-pr 5362

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2718 df-cleq 2731 df-clel 2814 df-ne 2935 df-ral 3054 df-rex 3064 df-rab 3392 df-v 3433 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-sn 4556 df-pr 4558 df-op 4562 df-br 5073 df-opab 5135 df-mpt 5154 df-id 5513 df-eprel 5518 df-xp 5624 df-rel 5625 df-cnv 5626 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-suc 6316 df-ec 8635 df-qmap 38813 df-adjliftmap 38822 df-sucmap 38829

This theorem is referenced by: (None)

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