dfsucmap2 - Mathbox for Peter Mazsa

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Theorem dfsucmap2 38963

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 38962	. 2 ⊢ SucMap = ( I AdjLiftMap V)
2		dmi 5897	. . . . . 6 ⊢ dom I = V
3	2	reseq2i 5962	. . . . 5 ⊢ (( I ∪ ^◡ E ) ↾ dom I ) = (( I ∪ ^◡ E ) ↾ V)
4	3	dmeqi 5880	. . . 4 ⊢ dom (( I ∪ ^◡ E ) ↾ dom I ) = dom (( I ∪ ^◡ E ) ↾ V)
5	3	eceq2i 8721	. . . 4 ⊢ [𝑚](( I ∪ ^◡ E ) ↾ dom I ) = [𝑚](( I ∪ ^◡ E ) ↾ V)
6	4, 5	mpteq12i 5197	. . 3 ⊢ (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ dom I ) ↦ [𝑚](( I ∪ ^◡ E ) ↾ dom I )) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ V) ↦ [𝑚](( I ∪ ^◡ E ) ↾ V))
7		dfadjliftmap 38955	. . 3 ⊢ ( I AdjLiftMap dom I ) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ dom I ) ↦ [𝑚](( I ∪ ^◡ E ) ↾ dom I ))
8		dfadjliftmap 38955	. . 3 ⊢ ( I AdjLiftMap V) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ V) ↦ [𝑚](( I ∪ ^◡ E ) ↾ V))
9	6, 7, 8	3eqtr4i 2795	. 2 ⊢ ( I AdjLiftMap dom I ) = ( I AdjLiftMap V)
10	1, 9	eqtr4i 2788	1 ⊢ SucMap = ( I AdjLiftMap dom I )

Colors of variables: wff setvar class

Syntax hints: = wceq 1560 Vcvv 3454 ∪ cun 3902 ↦ cmpt 5181 I cid 5541 E cep 5546 ^◡ccnv 5646 dom cdm 5647 ↾ cres 5649 [cec 8676 AdjLiftMap cadjliftmap 38675 SucMap csucmap 38677

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-12 2212 ax-ext 2734 ax-sep 5246 ax-pr 5390

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-clab 2741 df-cleq 2754 df-clel 2837 df-ne 2958 df-ral 3077 df-rex 3087 df-rab 3415 df-v 3456 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4481 df-sn 4583 df-pr 4585 df-op 4589 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5542 df-eprel 5547 df-xp 5653 df-rel 5654 df-cnv 5655 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-suc 6352 df-ec 8680 df-qmap 38945 df-adjliftmap 38954 df-sucmap 38961

This theorem is referenced by: (None)

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