dfsucmap2 - Mathbox for Peter Mazsa

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Theorem dfsucmap2 39533

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 39532	. 2 ⊢ SucMap = ( I AdjLiftMap V)
2		dmi 5904	. . . . . 6 ⊢ dom I = V
3	2	reseq2i 5968	. . . . 5 ⊢ (( I ∪ ^◡ E ) ↾ dom I ) = (( I ∪ ^◡ E ) ↾ V)
4	3	dmeqi 5887	. . . 4 ⊢ dom (( I ∪ ^◡ E ) ↾ dom I ) = dom (( I ∪ ^◡ E ) ↾ V)
5	3	eceq2i 8768	. . . 4 ⊢ [𝑚](( I ∪ ^◡ E ) ↾ dom I ) = [𝑚](( I ∪ ^◡ E ) ↾ V)
6	4, 5	mpteq12i 5201	. . 3 ⊢ (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ dom I ) ↦ [𝑚](( I ∪ ^◡ E ) ↾ dom I )) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ V) ↦ [𝑚](( I ∪ ^◡ E ) ↾ V))
7		dfadjliftmap 39525	. . 3 ⊢ ( I AdjLiftMap dom I ) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ dom I ) ↦ [𝑚](( I ∪ ^◡ E ) ↾ dom I ))
8		dfadjliftmap 39525	. . 3 ⊢ ( I AdjLiftMap V) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ V) ↦ [𝑚](( I ∪ ^◡ E ) ↾ V))
9	6, 7, 8	3eqtr4i 2793	. 2 ⊢ ( I AdjLiftMap dom I ) = ( I AdjLiftMap V)
10	1, 9	eqtr4i 2786	1 ⊢ SucMap = ( I AdjLiftMap dom I )

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1569 Vcvv 3450 ∪ cun 4148 ↦ cmpt 5185 I cid 5545 E cep 5550 ^◡ccnv 5650 dom cdm 5651 ↾ cres 5653 [cec 8723 AdjLiftMap cadjliftmap 39245 SucMap csucmap 39247

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-pr 5391

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-clab 2739 df-cleq 2752 df-clel 2835 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-ss 3897 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-sn 4585 df-pr 4587 df-op 4835 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5546 df-eprel 5551 df-xp 5657 df-rel 5658 df-cnv 5659 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-suc 6371 df-ec 8727 df-qmap 39515 df-adjliftmap 39524 df-sucmap 39531

This theorem is used by: (None)

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