Theorem newval 28254

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 6911	. . . 4 ⊢ (𝑥 = 𝐴 → (M‘𝑥) = (M‘𝐴))
2		fveq2 6911	. . . 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 28248	. . 3 ⊢ N = (𝑥 ∈ On ↦ ((M‘𝑥) ∖ (O‘𝑥)))
5		fvex 6924	. . . 4 ⊢ (M‘𝐴) ∈ V
6	5	difexi 5297	. . 3 ⊢ ((M‘𝐴) ∖ (O‘𝐴)) ∈ V
7	3, 4, 6	fvmpt 7019	. 2 ⊢ (𝐴 ∈ On → (N‘𝐴) = ((M‘𝐴) ∖ (O‘𝐴)))
8	4	fvmptndm 7051	. . 3 ⊢ (¬ 𝐴 ∈ On → (N‘𝐴) = ∅)
9		df-made 28246	. . . . . . . . 9 ⊢ M = recs((𝑓 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑓 × 𝒫 ∪ ran 𝑓))))
10	9	tfr1 8436	. . . . . . . 8 ⊢ M Fn On
11	10	fndmi 6668	. . . . . . 7 ⊢ dom M = On
12	11	eleq2i 2856	. . . . . 6 ⊢ (𝐴 ∈ dom M ↔ 𝐴 ∈ On)
13		ndmfv 6943	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom M → (M‘𝐴) = ∅)
14	12, 13	sylnbir 334	. . . . 5 ⊢ (¬ 𝐴 ∈ On → (M‘𝐴) = ∅)
15	14	difeq1d 4065	. . . 4 ⊢ (¬ 𝐴 ∈ On → ((M‘𝐴) ∖ (O‘𝐴)) = (∅ ∖ (O‘𝐴)))
16		0dif 4359	. . . 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))

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ∖ cdif 4048  ∅c0 4281  𝒫 cpw 4560  ∪ cuni 4870   ↦ cmpt 5190   × cxp 5654  dom cdm 5656  ran crn 5657   “ cima 5659  Oncon0 6383  ‘cfv 6561   |s ccuts 28178  Mcmade 28241  Ocold 28242  Ncnew 28243

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-made 28246  df-new 28248

This theorem is used by:  new0  28283  madeun  28303  newbday  28321