dfsucmap2 - Mathbox for Peter Mazsa

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Theorem dfsucmap2 38802

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 38801	. 2 ⊢ SucMap = ( I AdjLiftMap V)
2		dmi 5871	. . . . . 6 ⊢ dom I = V
3	2	reseq2i 5936	. . . . 5 ⊢ (( I ∪ ^◡ E ) ↾ dom I ) = (( I ∪ ^◡ E ) ↾ V)
4	3	dmeqi 5854	. . . 4 ⊢ dom (( I ∪ ^◡ E ) ↾ dom I ) = dom (( I ∪ ^◡ E ) ↾ V)
5	3	eceq2i 8680	. . . 4 ⊢ [𝑚](( I ∪ ^◡ E ) ↾ dom I ) = [𝑚](( I ∪ ^◡ E ) ↾ V)
6	4, 5	mpteq12i 5183	. . 3 ⊢ (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ dom I ) ↦ [𝑚](( I ∪ ^◡ E ) ↾ dom I )) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ V) ↦ [𝑚](( I ∪ ^◡ E ) ↾ V))
7		dfadjliftmap 38794	. . 3 ⊢ ( I AdjLiftMap dom I ) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ dom I ) ↦ [𝑚](( I ∪ ^◡ E ) ↾ dom I ))
8		dfadjliftmap 38794	. . 3 ⊢ ( I AdjLiftMap V) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ V) ↦ [𝑚](( I ∪ ^◡ E ) ↾ V))
9	6, 7, 8	3eqtr4i 2770	. 2 ⊢ ( I AdjLiftMap dom I ) = ( I AdjLiftMap V)
10	1, 9	eqtr4i 2763	1 ⊢ SucMap = ( I AdjLiftMap dom I )

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 Vcvv 3430 ∪ cun 3888 ↦ cmpt 5167 I cid 5519 E cep 5524 ^◡ccnv 5624 dom cdm 5625 ↾ cres 5627 [cec 8635 AdjLiftMap cadjliftmap 38514 SucMap csucmap 38516

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-pr 5371

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-clab 2716 df-cleq 2729 df-clel 2812 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-sn 4569 df-pr 4571 df-op 4575 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5520 df-eprel 5525 df-xp 5631 df-rel 5632 df-cnv 5633 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-suc 6324 df-ec 8639 df-qmap 38784 df-adjliftmap 38793 df-sucmap 38800

This theorem is referenced by: (None)

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