Theorem newval 27815

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 6876	. . . 4 ⊢ (𝑥 = 𝐴 → ( M ‘𝑥) = ( M ‘𝐴))
2		fveq2 6876	. . . 4 ⊢ (𝑥 = 𝐴 → ( O ‘𝑥) = ( O ‘𝐴))
3	1, 2	difeq12d 4102	. . 3 ⊢ (𝑥 = 𝐴 → (( M ‘𝑥) ∖ ( O ‘𝑥)) = (( M ‘𝐴) ∖ ( O ‘𝐴)))
4		df-new 27809	. . 3 ⊢ N = (𝑥 ∈ On ↦ (( M ‘𝑥) ∖ ( O ‘𝑥)))
5		fvex 6889	. . . 4 ⊢ ( M ‘𝐴) ∈ V
6	5	difexi 5300	. . 3 ⊢ (( M ‘𝐴) ∖ ( O ‘𝐴)) ∈ V
7	3, 4, 6	fvmpt 6986	. 2 ⊢ (𝐴 ∈ On → ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)))
8	4	fvmptndm 7017	. . 3 ⊢ (¬ 𝐴 ∈ On → ( N ‘𝐴) = ∅)
9		df-made 27807	. . . . . . . . 9 ⊢ M = recs((𝑓 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑓 × 𝒫 ∪ ran 𝑓))))
10	9	tfr1 8411	. . . . . . . 8 ⊢ M Fn On
11	10	fndmi 6642	. . . . . . 7 ⊢ dom M = On
12	11	eleq2i 2826	. . . . . 6 ⊢ (𝐴 ∈ dom M ↔ 𝐴 ∈ On)
13		ndmfv 6911	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom M → ( M ‘𝐴) = ∅)
14	12, 13	sylnbir 331	. . . . 5 ⊢ (¬ 𝐴 ∈ On → ( M ‘𝐴) = ∅)
15	14	difeq1d 4100	. . . 4 ⊢ (¬ 𝐴 ∈ On → (( M ‘𝐴) ∖ ( O ‘𝐴)) = (∅ ∖ ( O ‘𝐴)))
16		0dif 4380	. . . 4 ⊢ (∅ ∖ ( O ‘𝐴)) = ∅
17	15, 16	eqtrdi 2786	. . 3 ⊢ (¬ 𝐴 ∈ On → (( M ‘𝐴) ∖ ( O ‘𝐴)) = ∅)
18	8, 17	eqtr4d 2773	. 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 3459   ∖ cdif 3923  ∅c0 4308  𝒫 cpw 4575  ∪ cuni 4883   ↦ cmpt 5201   × cxp 5652  dom cdm 5654  ran crn 5655   “ cima 5657  Oncon0 6352  ‘cfv 6531   |s cscut 27746   M cmade 27802   O cold 27803   N cnew 27804

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-made 27807  df-new 27809

This theorem is referenced by:  new0  27838  madeun  27847  newbday  27865