Theorem madefi 27893

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 6832	. . 3 ⊢ (𝑥 = 𝑦 → ( M ‘𝑥) = ( M ‘𝑦))
2	1	eleq1d 2822	. 2 ⊢ (𝑥 = 𝑦 → (( M ‘𝑥) ∈ Fin ↔ ( M ‘𝑦) ∈ Fin))
3		fveq2 6832	. . 3 ⊢ (𝑥 = 𝐴 → ( M ‘𝑥) = ( M ‘𝐴))
4	3	eleq1d 2822	. 2 ⊢ (𝑥 = 𝐴 → (( M ‘𝑥) ∈ Fin ↔ ( M ‘𝐴) ∈ Fin))
5		nnon 7814	. . . . . 6 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On)
6		madeval 27812	. . . . . 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 27816	. . . . . . . . . . 11 ⊢ M :On⟶𝒫 No
10		ffun 6663	. . . . . . . . . . 11 ⊢ ( M :On⟶𝒫 No → Fun M )
11	9, 10	ax-mp 5	. . . . . . . . . 10 ⊢ Fun M
12		nnfi 9093	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → 𝑥 ∈ Fin)
13		imafi 9216	. . . . . . . . . 10 ⊢ ((Fun M ∧ 𝑥 ∈ Fin) → ( M “ 𝑥) ∈ Fin)
14	11, 12, 13	sylancr 588	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → ( M “ 𝑥) ∈ Fin)
15	14	adantr 480	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin) → ( M “ 𝑥) ∈ Fin)
16		onss 7730	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
17	5, 16	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ω → 𝑥 ⊆ On)
18	9	fdmi 6671	. . . . . . . . . . 11 ⊢ dom M = On
19	17, 18	sseqtrrdi 3964	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → 𝑥 ⊆ dom M )
20		funimass4 6896	. . . . . . . . . 10 ⊢ ((Fun M ∧ 𝑥 ⊆ dom M ) → (( M “ 𝑥) ⊆ Fin ↔ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin))
21	11, 19, 20	sylancr 588	. . . . . . . . 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 9245	. . . . . . . 8 ⊢ ((( M “ 𝑥) ∈ Fin ∧ ( M “ 𝑥) ⊆ Fin) → ∪ ( M “ 𝑥) ∈ Fin)
24	15, 22, 23	syl2anc 585	. . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin) → ∪ ( M “ 𝑥) ∈ Fin)
25		pwfi 9220	. . . . . . 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 9221	. . . . . 6 ⊢ ((𝒫 ∪ ( M “ 𝑥) ∈ Fin ∧ 𝒫 ∪ ( M “ 𝑥) ∈ Fin) → (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥)) ∈ Fin)
28	26, 26, 27	syl2anc 585	. . . . 5 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin) → (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥)) ∈ Fin)
29		vex 3434	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
30	29	funimaex 6578	. . . . . . . . . 10 ⊢ (Fun M → ( M “ 𝑥) ∈ V)
31	11, 30	ax-mp 5	. . . . . . . . 9 ⊢ ( M “ 𝑥) ∈ V
32	31	uniex 7686	. . . . . . . 8 ⊢ ∪ ( M “ 𝑥) ∈ V
33	32	pwex 5315	. . . . . . 7 ⊢ 𝒫 ∪ ( M “ 𝑥) ∈ V
34	33, 33	xpex 7698	. . . . . 6 ⊢ (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥)) ∈ V
35		cutsf 27772	. . . . . . 7 ⊢ |s : <<s ⟶ No
36		ffun 6663	. . . . . . 7 ⊢ ( |s : <<s ⟶ No → Fun |s )
37	35, 36	ax-mp 5	. . . . . 6 ⊢ Fun |s
38		imadomg 10445	. . . . . 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 9114	. . . . 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 587	. . . 4 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin) → ( |s “ (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥))) ∈ Fin)
42	8, 41	eqeltrd 2837	. . 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 7829	1 ⊢ (𝐴 ∈ ω → ( M ‘𝐴) ∈ Fin)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430   ⊆ wss 3890  𝒫 cpw 4542  ∪ cuni 4851   class class class wbr 5086   × cxp 5620  dom cdm 5622   “ cima 5625  Oncon0 6315  Fun wfun 6484  ⟶wf 6486  ‘cfv 6490  ωcom 7808   ≼ cdom 8882  Fincfn 8884   No csur 27591   <<s cslts 27737   |s ccuts 27739   M cmade 27802

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 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-ac2 10374

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-1o 8396  df-2o 8397  df-er 8634  df-map 8766  df-en 8885  df-dom 8886  df-fin 8888  df-card 9852  df-acn 9855  df-ac 10027  df-no 27594  df-lts 27595  df-bday 27596  df-slts 27738  df-cuts 27740  df-made 27807

This theorem is referenced by:  oldfi  27894