dfsucmap2 - Mathbox for Peter Mazsa

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Theorem dfsucmap2 39173

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 39172	. 2 ⊢ SucMap = ( I AdjLiftMap V)
2		dmi 5913	. . . . . 6 ⊢ dom I = V
3	2	reseq2i 5977	. . . . 5 ⊢ (( I ∪ ^◡ E ) ↾ dom I ) = (( I ∪ ^◡ E ) ↾ V)
4	3	dmeqi 5896	. . . 4 ⊢ dom (( I ∪ ^◡ E ) ↾ dom I ) = dom (( I ∪ ^◡ E ) ↾ V)
5	3	eceq2i 8743	. . . 4 ⊢ [𝑚](( I ∪ ^◡ E ) ↾ dom I ) = [𝑚](( I ∪ ^◡ E ) ↾ V)
6	4, 5	mpteq12i 5210	. . 3 ⊢ (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ dom I ) ↦ [𝑚](( I ∪ ^◡ E ) ↾ dom I )) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ V) ↦ [𝑚](( I ∪ ^◡ E ) ↾ V))
7		dfadjliftmap 39165	. . 3 ⊢ ( I AdjLiftMap dom I ) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ dom I ) ↦ [𝑚](( I ∪ ^◡ E ) ↾ dom I ))
8		dfadjliftmap 39165	. . 3 ⊢ ( I AdjLiftMap V) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ V) ↦ [𝑚](( I ∪ ^◡ E ) ↾ V))
9	6, 7, 8	3eqtr4i 2798	. 2 ⊢ ( I AdjLiftMap dom I ) = ( I AdjLiftMap V)
10	1, 9	eqtr4i 2791	1 ⊢ SucMap = ( I AdjLiftMap dom I )

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 Vcvv 3457 ∪ cun 3904 ↦ cmpt 5194 I cid 5557 E cep 5562 ^◡ccnv 5662 dom cdm 5663 ↾ cres 5665 [cec 8698 AdjLiftMap cadjliftmap 38885 SucMap csucmap 38887

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 2148 ax-9 2156 ax-10 2179 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-pr 5406

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 2744 df-cleq 2757 df-clel 2840 df-ne 2961 df-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-sn 4592 df-pr 4594 df-op 4598 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-eprel 5563 df-xp 5669 df-rel 5670 df-cnv 5671 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-suc 6370 df-ec 8702 df-qmap 39155 df-adjliftmap 39164 df-sucmap 39171

This theorem is used by: (None)

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