Theorem madefi 27824

Description: The made set of an ordinal natural is finite. (Contributed by Scott Fenton, 20-Aug-2025.)

Assertion

Ref	Expression
madefi	⊢ (𝐴 ∈ ω → ( M ‘𝐴) ∈ Fin)

Proof of Theorem madefi

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6858	. . 3 ⊢ (𝑥 = 𝑦 → ( M ‘𝑥) = ( M ‘𝑦))
2	1	eleq1d 2813	. 2 ⊢ (𝑥 = 𝑦 → (( M ‘𝑥) ∈ Fin ↔ ( M ‘𝑦) ∈ Fin))
3		fveq2 6858	. . 3 ⊢ (𝑥 = 𝐴 → ( M ‘𝑥) = ( M ‘𝐴))
4	3	eleq1d 2813	. 2 ⊢ (𝑥 = 𝐴 → (( M ‘𝑥) ∈ Fin ↔ ( M ‘𝐴) ∈ Fin))
5		nnon 7848	. . . . . 6 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On)
6		madeval 27760	. . . . . 6 ⊢ (𝑥 ∈ On → ( M ‘𝑥) = ( |s “ (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥))))
7	5, 6	syl 17	. . . . 5 ⊢ (𝑥 ∈ ω → ( M ‘𝑥) = ( |s “ (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥))))
8	7	adantr 480	. . . 4 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin) → ( M ‘𝑥) = ( |s “ (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥))))
9		madef 27764	. . . . . . . . . . 11 ⊢ M :On⟶𝒫 No
10		ffun 6691	. . . . . . . . . . 11 ⊢ ( M :On⟶𝒫 No → Fun M )
11	9, 10	ax-mp 5	. . . . . . . . . 10 ⊢ Fun M
12		nnfi 9131	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → 𝑥 ∈ Fin)
13		imafi 9264	. . . . . . . . . 10 ⊢ ((Fun M ∧ 𝑥 ∈ Fin) → ( M “ 𝑥) ∈ Fin)
14	11, 12, 13	sylancr 587	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → ( M “ 𝑥) ∈ Fin)
15	14	adantr 480	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin) → ( M “ 𝑥) ∈ Fin)
16		onss 7761	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
17	5, 16	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ω → 𝑥 ⊆ On)
18	9	fdmi 6699	. . . . . . . . . . 11 ⊢ dom M = On
19	17, 18	sseqtrrdi 3988	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → 𝑥 ⊆ dom M )
20		funimass4 6925	. . . . . . . . . 10 ⊢ ((Fun M ∧ 𝑥 ⊆ dom M ) → (( M “ 𝑥) ⊆ Fin ↔ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin))
21	11, 19, 20	sylancr 587	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → (( M “ 𝑥) ⊆ Fin ↔ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin))
22	21	biimpar 477	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin) → ( M “ 𝑥) ⊆ Fin)
23		unifi 9295	. . . . . . . 8 ⊢ ((( M “ 𝑥) ∈ Fin ∧ ( M “ 𝑥) ⊆ Fin) → ∪ ( M “ 𝑥) ∈ Fin)
24	15, 22, 23	syl2anc 584	. . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin) → ∪ ( M “ 𝑥) ∈ Fin)
25		pwfi 9268	. . . . . . 7 ⊢ (∪ ( M “ 𝑥) ∈ Fin ↔ 𝒫 ∪ ( M “ 𝑥) ∈ Fin)
26	24, 25	sylib 218	. . . . . 6 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin) → 𝒫 ∪ ( M “ 𝑥) ∈ Fin)
27		xpfi 9269	. . . . . 6 ⊢ ((𝒫 ∪ ( M “ 𝑥) ∈ Fin ∧ 𝒫 ∪ ( M “ 𝑥) ∈ Fin) → (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥)) ∈ Fin)
28	26, 26, 27	syl2anc 584	. . . . 5 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin) → (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥)) ∈ Fin)
29		vex 3451	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
30	29	funimaex 6605	. . . . . . . . . 10 ⊢ (Fun M → ( M “ 𝑥) ∈ V)
31	11, 30	ax-mp 5	. . . . . . . . 9 ⊢ ( M “ 𝑥) ∈ V
32	31	uniex 7717	. . . . . . . 8 ⊢ ∪ ( M “ 𝑥) ∈ V
33	32	pwex 5335	. . . . . . 7 ⊢ 𝒫 ∪ ( M “ 𝑥) ∈ V
34	33, 33	xpex 7729	. . . . . 6 ⊢ (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥)) ∈ V
35		scutf 27724	. . . . . . 7 ⊢ |s : <<s ⟶ No
36		ffun 6691	. . . . . . 7 ⊢ ( |s : <<s ⟶ No → Fun |s )
37	35, 36	ax-mp 5	. . . . . 6 ⊢ Fun |s
38		imadomg 10487	. . . . . 6 ⊢ ((𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥)) ∈ V → (Fun |s → ( |s “ (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥))) ≼ (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥))))
39	34, 37, 38	mp2 9	. . . . 5 ⊢ ( |s “ (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥))) ≼ (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥))
40		domfi 9153	. . . . 5 ⊢ (((𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥)) ∈ Fin ∧ ( |s “ (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥))) ≼ (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥))) → ( |s “ (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥))) ∈ Fin)
41	28, 39, 40	sylancl 586	. . . 4 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin) → ( |s “ (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥))) ∈ Fin)
42	8, 41	eqeltrd 2828	. . 3 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin) → ( M ‘𝑥) ∈ Fin)
43	42	ex 412	. 2 ⊢ (𝑥 ∈ ω → (∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin → ( M ‘𝑥) ∈ Fin))
44	2, 4, 43	omsinds 7863	1 ⊢ (𝐴 ∈ ω → ( M ‘𝐴) ∈ Fin)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3447   ⊆ wss 3914  𝒫 cpw 4563  ∪ cuni 4871   class class class wbr 5107   × cxp 5636  dom cdm 5638   “ cima 5641  Oncon0 6332  Fun wfun 6505  ⟶wf 6507  ‘cfv 6511  ωcom 7842   ≼ cdom 8916  Fincfn 8918   No csur 27551   <<s csslt 27692   |s cscut 27694   M cmade 27750

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-ac2 10416

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-1o 8434  df-2o 8435  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-fin 8922  df-card 9892  df-acn 9895  df-ac 10069  df-no 27554  df-slt 27555  df-bday 27556  df-sslt 27693  df-scut 27695  df-made 27755

This theorem is referenced by:  oldfi  27825