dfsucmap2 - Mathbox for Peter Mazsa

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Theorem dfsucmap2 38487

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 38486	. 2 ⊢ SucMap = ( I AdjLiftMap V)
2		dmi 5860	. . . . . 6 ⊢ dom I = V
3	2	reseq2i 5924	. . . . 5 ⊢ (( I ∪ ^◡ E ) ↾ dom I ) = (( I ∪ ^◡ E ) ↾ V)
4	3	dmeqi 5843	. . . 4 ⊢ dom (( I ∪ ^◡ E ) ↾ dom I ) = dom (( I ∪ ^◡ E ) ↾ V)
5	3	eceq2i 8664	. . . 4 ⊢ [𝑚](( I ∪ ^◡ E ) ↾ dom I ) = [𝑚](( I ∪ ^◡ E ) ↾ V)
6	4, 5	mpteq12i 5186	. . 3 ⊢ (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ dom I ) ↦ [𝑚](( I ∪ ^◡ E ) ↾ dom I )) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ V) ↦ [𝑚](( I ∪ ^◡ E ) ↾ V))
7		df-adjliftmap 38480	. . 3 ⊢ ( I AdjLiftMap dom I ) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ dom I ) ↦ [𝑚](( I ∪ ^◡ E ) ↾ dom I ))
8		df-adjliftmap 38480	. . 3 ⊢ ( I AdjLiftMap V) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ V) ↦ [𝑚](( I ∪ ^◡ E ) ↾ V))
9	6, 7, 8	3eqtr4i 2764	. 2 ⊢ ( I AdjLiftMap dom I ) = ( I AdjLiftMap V)
10	1, 9	eqtr4i 2757	1 ⊢ SucMap = ( I AdjLiftMap dom I )

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 Vcvv 3436 ∪ cun 3895 ↦ cmpt 5170 I cid 5508 E cep 5513 ^◡ccnv 5613 dom cdm 5614 ↾ cres 5616 [cec 8620 AdjLiftMap cadjliftmap 38225 SucMap csucmap 38227

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-10 2144 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-nf 1785 df-sb 2068 df-clab 2710 df-cleq 2723 df-clel 2806 df-ne 2929 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-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-eprel 5514 df-xp 5620 df-rel 5621 df-cnv 5622 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-suc 6312 df-ec 8624 df-adjliftmap 38480 df-sucmap 38485

This theorem is referenced by: (None)

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