Theorem madefi 27926

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 6844	. . 3 ⊢ (𝑥 = 𝑦 → ( M ‘𝑥) = ( M ‘𝑦))
2	1	eleq1d 2822	. 2 ⊢ (𝑥 = 𝑦 → (( M ‘𝑥) ∈ Fin ↔ ( M ‘𝑦) ∈ Fin))
3		fveq2 6844	. . 3 ⊢ (𝑥 = 𝐴 → ( M ‘𝑥) = ( M ‘𝐴))
4	3	eleq1d 2822	. 2 ⊢ (𝑥 = 𝐴 → (( M ‘𝑥) ∈ Fin ↔ ( M ‘𝐴) ∈ Fin))
5		nnon 7826	. . . . . 6 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On)
6		madeval 27845	. . . . . 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 27849	. . . . . . . . . . 11 ⊢ M :On⟶𝒫 No
10		ffun 6675	. . . . . . . . . . 11 ⊢ ( M :On⟶𝒫 No → Fun M )
11	9, 10	ax-mp 5	. . . . . . . . . 10 ⊢ Fun M
12		nnfi 9106	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → 𝑥 ∈ Fin)
13		imafi 9229	. . . . . . . . . 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 7742	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
17	5, 16	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ω → 𝑥 ⊆ On)
18	9	fdmi 6683	. . . . . . . . . . 11 ⊢ dom M = On
19	17, 18	sseqtrrdi 3977	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → 𝑥 ⊆ dom M )
20		funimass4 6908	. . . . . . . . . 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 9258	. . . . . . . 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 9233	. . . . . . 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 9234	. . . . . 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 3446	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
30	29	funimaex 6590	. . . . . . . . . 10 ⊢ (Fun M → ( M “ 𝑥) ∈ V)
31	11, 30	ax-mp 5	. . . . . . . . 9 ⊢ ( M “ 𝑥) ∈ V
32	31	uniex 7698	. . . . . . . 8 ⊢ ∪ ( M “ 𝑥) ∈ V
33	32	pwex 5329	. . . . . . 7 ⊢ 𝒫 ∪ ( M “ 𝑥) ∈ V
34	33, 33	xpex 7710	. . . . . 6 ⊢ (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥)) ∈ V
35		cutsf 27805	. . . . . . 7 ⊢ |s : <<s ⟶ No
36		ffun 6675	. . . . . . 7 ⊢ ( |s : <<s ⟶ No → Fun |s )
37	35, 36	ax-mp 5	. . . . . 6 ⊢ Fun |s
38		imadomg 10458	. . . . . 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 9127	. . . . 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 7841	1 ⊢ (𝐴 ∈ ω → ( M ‘𝐴) ∈ Fin)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3442   ⊆ wss 3903  𝒫 cpw 4556  ∪ cuni 4865   class class class wbr 5100   × cxp 5632  dom cdm 5634   “ cima 5637  Oncon0 6327  Fun wfun 6496  ⟶wf 6498  ‘cfv 6502  ωcom 7820   ≼ cdom 8895  Fincfn 8897   No csur 27624   <<s cslts 27770   |s ccuts 27772   M cmade 27835

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 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-ac2 10387

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 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-se 5588  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-isom 6511  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-1o 8409  df-2o 8410  df-er 8647  df-map 8779  df-en 8898  df-dom 8899  df-fin 8901  df-card 9865  df-acn 9868  df-ac 10040  df-no 27627  df-lts 27628  df-bday 27629  df-slts 27771  df-cuts 27773  df-made 27840

This theorem is referenced by:  oldfi  27927