Theorem newval 28009

Description: The value of the new options function. (Contributed by Scott Fenton, 9-Oct-2024.)

Assertion

Ref	Expression
newval	⊢ ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))

Proof of Theorem newval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6883	. . . 4 ⊢ (𝑥 = 𝐴 → ( M ‘𝑥) = ( M ‘𝐴))
2		fveq2 6883	. . . 4 ⊢ (𝑥 = 𝐴 → ( O ‘𝑥) = ( O ‘𝐴))
3	1, 2	difeq12d 4083	. . 3 ⊢ (𝑥 = 𝐴 → (( M ‘𝑥) ∖ ( O ‘𝑥)) = (( M ‘𝐴) ∖ ( O ‘𝐴)))
4		df-new 28003	. . 3 ⊢ N = (𝑥 ∈ On ↦ (( M ‘𝑥) ∖ ( O ‘𝑥)))
5		fvex 6896	. . . 4 ⊢ ( M ‘𝐴) ∈ V
6	5	difexi 5302	. . 3 ⊢ (( M ‘𝐴) ∖ ( O ‘𝐴)) ∈ V
7	3, 4, 6	fvmpt 6991	. 2 ⊢ (𝐴 ∈ On → ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)))
8	4	fvmptndm 7023	. . 3 ⊢ (¬ 𝐴 ∈ On → ( N ‘𝐴) = ∅)
9		df-made 28001	. . . . . . . . 9 ⊢ M = recs((𝑓 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑓 × 𝒫 ∪ ran 𝑓))))
10	9	tfr1 8385	. . . . . . . 8 ⊢ M Fn On
11	10	fndmi 6641	. . . . . . 7 ⊢ dom M = On
12	11	eleq2i 2855	. . . . . 6 ⊢ (𝐴 ∈ dom M ↔ 𝐴 ∈ On)
13		ndmfv 6915	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom M → ( M ‘𝐴) = ∅)
14	12, 13	sylnbir 334	. . . . 5 ⊢ (¬ 𝐴 ∈ On → ( M ‘𝐴) = ∅)
15	14	difeq1d 4081	. . . 4 ⊢ (¬ 𝐴 ∈ On → (( M ‘𝐴) ∖ ( O ‘𝐴)) = (∅ ∖ ( O ‘𝐴)))
16		0dif 4364	. . . 4 ⊢ (∅ ∖ ( O ‘𝐴)) = ∅
17	15, 16	eqtrdi 2814	. . 3 ⊢ (¬ 𝐴 ∈ On → (( M ‘𝐴) ∖ ( O ‘𝐴)) = ∅)
18	8, 17	eqtr4d 2801	. 2 ⊢ (¬ 𝐴 ∈ On → ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)))
19	7, 18	pm2.61i 184	1 ⊢ ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))

Syntax hints:  ¬ wn 3   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∖ cdif 3903  ∅c0 4287  𝒫 cpw 4563  ∪ cuni 4873   ↦ cmpt 5193   × cxp 5661  dom cdm 5663  ran crn 5664   “ cima 5666  Oncon0 6362  ‘cfv 6538   |s ccuts 27933   M cmade 27996   O cold 27997   N cnew 27998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-made 28001  df-new 28003

This theorem is referenced by:  new0  28038  madeun  28058  newbday  28076