Theorem newval 28227

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 6886	. . . 4 ⊢ (𝑥 = 𝐴 → (M‘𝑥) = (M‘𝐴))
2		fveq2 6886	. . . 4 ⊢ (𝑥 = 𝐴 → (O‘𝑥) = (O‘𝐴))
3	1, 2	difeq12d 4064	. . 3 ⊢ (𝑥 = 𝐴 → ((M‘𝑥) ∖ (O‘𝑥)) = ((M‘𝐴) ∖ (O‘𝐴)))
4		df-new 28221	. . 3 ⊢ N = (𝑥 ∈ On ↦ ((M‘𝑥) ∖ (O‘𝑥)))
5		fvex 6899	. . . 4 ⊢ (M‘𝐴) ∈ V
6	5	difexi 5291	. . 3 ⊢ ((M‘𝐴) ∖ (O‘𝐴)) ∈ V
7	3, 4, 6	fvmpt 6994	. 2 ⊢ (𝐴 ∈ On → (N‘𝐴) = ((M‘𝐴) ∖ (O‘𝐴)))
8	4	fvmptndm 7026	. . 3 ⊢ (¬ 𝐴 ∈ On → (N‘𝐴) = ∅)
9		df-made 28219	. . . . . . . . 9 ⊢ M = recs((𝑓 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑓 × 𝒫 ∪ ran 𝑓))))
10	9	tfr1 8409	. . . . . . . 8 ⊢ M Fn On
11	10	fndmi 6644	. . . . . . 7 ⊢ dom M = On
12	11	eleq2i 2852	. . . . . 6 ⊢ (𝐴 ∈ dom M ↔ 𝐴 ∈ On)
13		ndmfv 6918	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom M → (M‘𝐴) = ∅)
14	12, 13	sylnbir 334	. . . . 5 ⊢ (¬ 𝐴 ∈ On → (M‘𝐴) = ∅)
15	14	difeq1d 4062	. . . 4 ⊢ (¬ 𝐴 ∈ On → ((M‘𝐴) ∖ (O‘𝐴)) = (∅ ∖ (O‘𝐴)))
16		0dif 4355	. . . 4 ⊢ (∅ ∖ (O‘𝐴)) = ∅
17	15, 16	eqtrdi 2811	. . 3 ⊢ (¬ 𝐴 ∈ On → ((M‘𝐴) ∖ (O‘𝐴)) = ∅)
18	8, 17	eqtr4d 2798	. 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 1569   ∈ wcel 2144  Vcvv 3450   ∖ cdif 4045  ∅c0 4278  𝒫 cpw 4556  ∪ cuni 4866   ↦ cmpt 5185   × cxp 5648  dom cdm 5650  ran crn 5651   “ cima 5653  Oncon0 6362  ‘cfv 6539   |s ccuts 28151  Mcmade 28214  Ocold 28215  Ncnew 28216

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-made 28219  df-new 28221

This theorem is used by:  new0  28256  madeun  28276  newbday  28294