dfsucmap2 - Mathbox for Peter Mazsa

	Mathbox for Peter Mazsa		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dfsucmap2		Structured version Visualization version GIF version

Theorem dfsucmap2 38577

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 38576	. 2 ⊢ SucMap = ( I AdjLiftMap V)
2		dmi 5868	. . . . . 6 ⊢ dom I = V
3	2	reseq2i 5933	. . . . 5 ⊢ (( I ∪ ^◡ E ) ↾ dom I ) = (( I ∪ ^◡ E ) ↾ V)
4	3	dmeqi 5851	. . . 4 ⊢ dom (( I ∪ ^◡ E ) ↾ dom I ) = dom (( I ∪ ^◡ E ) ↾ V)
5	3	eceq2i 8675	. . . 4 ⊢ [𝑚](( I ∪ ^◡ E ) ↾ dom I ) = [𝑚](( I ∪ ^◡ E ) ↾ V)
6	4, 5	mpteq12i 5193	. . 3 ⊢ (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ dom I ) ↦ [𝑚](( I ∪ ^◡ E ) ↾ dom I )) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ V) ↦ [𝑚](( I ∪ ^◡ E ) ↾ V))
7		df-adjliftmap 38570	. . 3 ⊢ ( I AdjLiftMap dom I ) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ dom I ) ↦ [𝑚](( I ∪ ^◡ E ) ↾ dom I ))
8		df-adjliftmap 38570	. . 3 ⊢ ( I AdjLiftMap V) = (𝑚 ∈ dom (( I ∪ ^◡ E ) ↾ V) ↦ [𝑚](( I ∪ ^◡ E ) ↾ V))
9	6, 7, 8	3eqtr4i 2767	. 2 ⊢ ( I AdjLiftMap dom I ) = ( I AdjLiftMap V)
10	1, 9	eqtr4i 2760	1 ⊢ SucMap = ( I AdjLiftMap dom I )

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 Vcvv 3438 ∪ cun 3897 ↦ cmpt 5177 I cid 5516 E cep 5521 ^◡ccnv 5621 dom cdm 5622 ↾ cres 5624 [cec 8631 AdjLiftMap cadjliftmap 38315 SucMap csucmap 38317

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 2115 ax-9 2123 ax-10 2146 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pr 5375

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 2713 df-cleq 2726 df-clel 2809 df-ne 2931 df-ral 3050 df-rex 3059 df-rab 3398 df-v 3440 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-sn 4579 df-pr 4581 df-op 4585 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-eprel 5522 df-xp 5628 df-rel 5629 df-cnv 5630 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-suc 6321 df-ec 8635 df-adjliftmap 38570 df-sucmap 38575

This theorem is referenced by: (None)

W3C validator