Theorem newval 27852

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 6834	. . . 4 ⊢ (𝑥 = 𝐴 → ( M ‘𝑥) = ( M ‘𝐴))
2		fveq2 6834	. . . 4 ⊢ (𝑥 = 𝐴 → ( O ‘𝑥) = ( O ‘𝐴))
3	1, 2	difeq12d 4065	. . 3 ⊢ (𝑥 = 𝐴 → (( M ‘𝑥) ∖ ( O ‘𝑥)) = (( M ‘𝐴) ∖ ( O ‘𝐴)))
4		df-new 27846	. . 3 ⊢ N = (𝑥 ∈ On ↦ (( M ‘𝑥) ∖ ( O ‘𝑥)))
5		fvex 6847	. . . 4 ⊢ ( M ‘𝐴) ∈ V
6	5	difexi 5265	. . 3 ⊢ (( M ‘𝐴) ∖ ( O ‘𝐴)) ∈ V
7	3, 4, 6	fvmpt 6942	. 2 ⊢ (𝐴 ∈ On → ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)))
8	4	fvmptndm 6974	. . 3 ⊢ (¬ 𝐴 ∈ On → ( N ‘𝐴) = ∅)
9		df-made 27844	. . . . . . . . 9 ⊢ M = recs((𝑓 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑓 × 𝒫 ∪ ran 𝑓))))
10	9	tfr1 8333	. . . . . . . 8 ⊢ M Fn On
11	10	fndmi 6596	. . . . . . 7 ⊢ dom M = On
12	11	eleq2i 2832	. . . . . 6 ⊢ (𝐴 ∈ dom M ↔ 𝐴 ∈ On)
13		ndmfv 6866	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom M → ( M ‘𝐴) = ∅)
14	12, 13	sylnbir 332	. . . . 5 ⊢ (¬ 𝐴 ∈ On → ( M ‘𝐴) = ∅)
15	14	difeq1d 4063	. . . 4 ⊢ (¬ 𝐴 ∈ On → (( M ‘𝐴) ∖ ( O ‘𝐴)) = (∅ ∖ ( O ‘𝐴)))
16		0dif 4340	. . . 4 ⊢ (∅ ∖ ( O ‘𝐴)) = ∅
17	15, 16	eqtrdi 2791	. . 3 ⊢ (¬ 𝐴 ∈ On → (( M ‘𝐴) ∖ ( O ‘𝐴)) = ∅)
18	8, 17	eqtr4d 2778	. 2 ⊢ (¬ 𝐴 ∈ On → ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)))
19	7, 18	pm2.61i 183	1 ⊢ ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))

Syntax hints:  ¬ wn 3   = wceq 1547   ∈ wcel 2119  Vcvv 3432   ∖ cdif 3887  ∅c0 4268  𝒫 cpw 4536  ∪ cuni 4845   ↦ cmpt 5160   × cxp 5623  dom cdm 5625  ran crn 5626   “ cima 5628  Oncon0 6317  ‘cfv 6492   |s ccuts 27776   M cmade 27839   O cold 27840   N cnew 27841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-made 27844  df-new 27846

This theorem is referenced by:  new0  27881  madeun  27901  newbday  27919