dfsucmap2 - Mathbox for Peter Mazsa

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Theorem dfsucmap2 39396

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 39395	. 2 ⊢ SucMap = ( I AdjLiftMap V)
2		dmi 5903	. . . . . 6 ⊢ dom I = V
3	2	reseq2i 5967	. . . . 5 ⊢ (( I ∪ ^◡ E ) ↾ dom I ) = (( I ∪ ^◡ E ) ↾ V)
4	3	dmeqi 5886	. . . 4 ⊢ dom (( I ∪ ^◡ E ) ↾ dom I ) = dom (( I ∪ ^◡ E ) ↾ V)
5	3	eceq2i 8760	. . . 4 ⊢ [𝑚](( I ∪ ^◡ E ) ↾ dom I ) = [𝑚](( I ∪ ^◡ E ) ↾ V)
6	4, 5	mpteq12i 5202	. . 3 ⊢ (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ dom I ) ↦ [𝑚](( I ∪ ^◡ E ) ↾ dom I )) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ V) ↦ [𝑚](( I ∪ ^◡ E ) ↾ V))
7		dfadjliftmap 39388	. . 3 ⊢ ( I AdjLiftMap dom I ) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ dom I ) ↦ [𝑚](( I ∪ ^◡ E ) ↾ dom I ))
8		dfadjliftmap 39388	. . 3 ⊢ ( I AdjLiftMap V) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ V) ↦ [𝑚](( I ∪ ^◡ E ) ↾ V))
9	6, 7, 8	3eqtr4i 2794	. 2 ⊢ ( I AdjLiftMap dom I ) = ( I AdjLiftMap V)
10	1, 9	eqtr4i 2787	1 ⊢ SucMap = ( I AdjLiftMap dom I )

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 Vcvv 3451 ∪ cun 3897 ↦ cmpt 5186 I cid 5545 E cep 5550 ^◡ccnv 5650 dom cdm 5651 ↾ cres 5653 [cec 8715 AdjLiftMap cadjliftmap 39108 SucMap csucmap 39110

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-pr 5391

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-clab 2740 df-cleq 2753 df-clel 2836 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5168 df-mpt 5187 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 6368 df-ec 8719 df-qmap 39378 df-adjliftmap 39387 df-sucmap 39394

This theorem is used by: (None)

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