Theorem newval 27827

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 6840	. . . 4 ⊢ (𝑥 = 𝐴 → ( M ‘𝑥) = ( M ‘𝐴))
2		fveq2 6840	. . . 4 ⊢ (𝑥 = 𝐴 → ( O ‘𝑥) = ( O ‘𝐴))
3	1, 2	difeq12d 4067	. . 3 ⊢ (𝑥 = 𝐴 → (( M ‘𝑥) ∖ ( O ‘𝑥)) = (( M ‘𝐴) ∖ ( O ‘𝐴)))
4		df-new 27821	. . 3 ⊢ N = (𝑥 ∈ On ↦ (( M ‘𝑥) ∖ ( O ‘𝑥)))
5		fvex 6853	. . . 4 ⊢ ( M ‘𝐴) ∈ V
6	5	difexi 5271	. . 3 ⊢ (( M ‘𝐴) ∖ ( O ‘𝐴)) ∈ V
7	3, 4, 6	fvmpt 6947	. 2 ⊢ (𝐴 ∈ On → ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)))
8	4	fvmptndm 6979	. . 3 ⊢ (¬ 𝐴 ∈ On → ( N ‘𝐴) = ∅)
9		df-made 27819	. . . . . . . . 9 ⊢ M = recs((𝑓 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑓 × 𝒫 ∪ ran 𝑓))))
10	9	tfr1 8336	. . . . . . . 8 ⊢ M Fn On
11	10	fndmi 6602	. . . . . . 7 ⊢ dom M = On
12	11	eleq2i 2828	. . . . . 6 ⊢ (𝐴 ∈ dom M ↔ 𝐴 ∈ On)
13		ndmfv 6872	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom M → ( M ‘𝐴) = ∅)
14	12, 13	sylnbir 331	. . . . 5 ⊢ (¬ 𝐴 ∈ On → ( M ‘𝐴) = ∅)
15	14	difeq1d 4065	. . . 4 ⊢ (¬ 𝐴 ∈ On → (( M ‘𝐴) ∖ ( O ‘𝐴)) = (∅ ∖ ( O ‘𝐴)))
16		0dif 4345	. . . 4 ⊢ (∅ ∖ ( O ‘𝐴)) = ∅
17	15, 16	eqtrdi 2787	. . 3 ⊢ (¬ 𝐴 ∈ On → (( M ‘𝐴) ∖ ( O ‘𝐴)) = ∅)
18	8, 17	eqtr4d 2774	. 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 1542   ∈ wcel 2114  Vcvv 3429   ∖ cdif 3886  ∅c0 4273  𝒫 cpw 4541  ∪ cuni 4850   ↦ cmpt 5166   × cxp 5629  dom cdm 5631  ran crn 5632   “ cima 5634  Oncon0 6323  ‘cfv 6498   |s ccuts 27751   M cmade 27814   O cold 27815   N cnew 27816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-made 27819  df-new 27821

This theorem is referenced by:  new0  27856  madeun  27876  newbday  27894