Theorem newval 27894

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 6906	. . . 4 ⊢ (𝑥 = 𝐴 → ( M ‘𝑥) = ( M ‘𝐴))
2		fveq2 6906	. . . 4 ⊢ (𝑥 = 𝐴 → ( O ‘𝑥) = ( O ‘𝐴))
3	1, 2	difeq12d 4127	. . 3 ⊢ (𝑥 = 𝐴 → (( M ‘𝑥) ∖ ( O ‘𝑥)) = (( M ‘𝐴) ∖ ( O ‘𝐴)))
4		df-new 27888	. . 3 ⊢ N = (𝑥 ∈ On ↦ (( M ‘𝑥) ∖ ( O ‘𝑥)))
5		fvex 6919	. . . 4 ⊢ ( M ‘𝐴) ∈ V
6	5	difexi 5330	. . 3 ⊢ (( M ‘𝐴) ∖ ( O ‘𝐴)) ∈ V
7	3, 4, 6	fvmpt 7016	. 2 ⊢ (𝐴 ∈ On → ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)))
8	4	fvmptndm 7047	. . 3 ⊢ (¬ 𝐴 ∈ On → ( N ‘𝐴) = ∅)
9		df-made 27886	. . . . . . . . 9 ⊢ M = recs((𝑓 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑓 × 𝒫 ∪ ran 𝑓))))
10	9	tfr1 8437	. . . . . . . 8 ⊢ M Fn On
11	10	fndmi 6672	. . . . . . 7 ⊢ dom M = On
12	11	eleq2i 2833	. . . . . 6 ⊢ (𝐴 ∈ dom M ↔ 𝐴 ∈ On)
13		ndmfv 6941	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom M → ( M ‘𝐴) = ∅)
14	12, 13	sylnbir 331	. . . . 5 ⊢ (¬ 𝐴 ∈ On → ( M ‘𝐴) = ∅)
15	14	difeq1d 4125	. . . 4 ⊢ (¬ 𝐴 ∈ On → (( M ‘𝐴) ∖ ( O ‘𝐴)) = (∅ ∖ ( O ‘𝐴)))
16		0dif 4405	. . . 4 ⊢ (∅ ∖ ( O ‘𝐴)) = ∅
17	15, 16	eqtrdi 2793	. . 3 ⊢ (¬ 𝐴 ∈ On → (( M ‘𝐴) ∖ ( O ‘𝐴)) = ∅)
18	8, 17	eqtr4d 2780	. 2 ⊢ (¬ 𝐴 ∈ On → ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)))
19	7, 18	pm2.61i 182	1 ⊢ ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2108  Vcvv 3480   ∖ cdif 3948  ∅c0 4333  𝒫 cpw 4600  ∪ cuni 4907   ↦ cmpt 5225   × cxp 5683  dom cdm 5685  ran crn 5686   “ cima 5688  Oncon0 6384  ‘cfv 6561   |s cscut 27827   M cmade 27881   O cold 27882   N cnew 27883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-made 27886  df-new 27888

This theorem is referenced by:  new0  27913  madeun  27922  newbday  27940