Theorem madefi 28299

Description: The made set of an ordinal natural is finite. (Contributed by Scott Fenton, 20-Aug-2025.) (Proof shortened by Vincent Gonzalez, 19-Aug-2026.)

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 6885	. . 3 ⊢ (𝑥 = 𝑦 → (M‘𝑥) = (M‘𝑦))
2	1	eleq1d 2845	. 2 ⊢ (𝑥 = 𝑦 → ((M‘𝑥) ∈ Fin ↔ (M‘𝑦) ∈ Fin))
3		fveq2 6885	. . 3 ⊢ (𝑥 = 𝐴 → (M‘𝑥) = (M‘𝐴))
4	3	eleq1d 2845	. 2 ⊢ (𝑥 = 𝐴 → ((M‘𝑥) ∈ Fin ↔ (M‘𝐴) ∈ Fin))
5		nnon 7883	. . . . . 6 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On)
6		madeval 28218	. . . . . 6 ⊢ (𝑥 ∈ On → (M‘𝑥) = ( |s “ (𝒫 ∪ (M “ 𝑥) × 𝒫 ∪ (M “ 𝑥))))
7	5, 6	syl 18	. . . . 5 ⊢ (𝑥 ∈ ω → (M‘𝑥) = ( |s “ (𝒫 ∪ (M “ 𝑥) × 𝒫 ∪ (M “ 𝑥))))
8	7	adantr 486	. . . 4 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 (M‘𝑦) ∈ Fin) → (M‘𝑥) = ( |s “ (𝒫 ∪ (M “ 𝑥) × 𝒫 ∪ (M “ 𝑥))))
9		cutsf 28178	. . . . . 6 ⊢ |s : <<s ⟶No
10		ffun 6712	. . . . . 6 ⊢ ( |s : <<s ⟶No → Fun |s )
11	9, 10	ax-mp 5	. . . . 5 ⊢ Fun |s
12		madef 28222	. . . . . . . . . . 11 ⊢ M:On⟶𝒫 No
13		ffun 6712	. . . . . . . . . . 11 ⊢ (M:On⟶𝒫 No → Fun M)
14	12, 13	ax-mp 5	. . . . . . . . . 10 ⊢ Fun M
15		nnfi 9183	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → 𝑥 ∈ Fin)
16		imafi 9307	. . . . . . . . . 10 ⊢ ((Fun M ∧ 𝑥 ∈ Fin) → (M “ 𝑥) ∈ Fin)
17	14, 15, 16	sylancr 599	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → (M “ 𝑥) ∈ Fin)
18	17	adantr 486	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 (M‘𝑦) ∈ Fin) → (M “ 𝑥) ∈ Fin)
19		onss 7799	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
20	5, 19	syl 18	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ω → 𝑥 ⊆ On)
21	12	fdmi 6721	. . . . . . . . . . 11 ⊢ dom M = On
22	20, 21	sseqtrrdi 3949	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → 𝑥 ⊆ dom M)
23		funimass4 6949	. . . . . . . . . 10 ⊢ ((Fun M ∧ 𝑥 ⊆ dom M) → ((M “ 𝑥) ⊆ Fin ↔ ∀𝑦 ∈ 𝑥 (M‘𝑦) ∈ Fin))
24	14, 22, 23	sylancr 599	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → ((M “ 𝑥) ⊆ Fin ↔ ∀𝑦 ∈ 𝑥 (M‘𝑦) ∈ Fin))
25	24	biimpar 483	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 (M‘𝑦) ∈ Fin) → (M “ 𝑥) ⊆ Fin)
26		unifi 9333	. . . . . . . 8 ⊢ (((M “ 𝑥) ∈ Fin ∧ (M “ 𝑥) ⊆ Fin) → ∪ (M “ 𝑥) ∈ Fin)
27	18, 25, 26	syl2anc 596	. . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 (M‘𝑦) ∈ Fin) → ∪ (M “ 𝑥) ∈ Fin)
28		pwfi 9310	. . . . . . 7 ⊢ (∪ (M “ 𝑥) ∈ Fin ↔ 𝒫 ∪ (M “ 𝑥) ∈ Fin)
29	27, 28	sylib 221	. . . . . 6 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 (M‘𝑦) ∈ Fin) → 𝒫 ∪ (M “ 𝑥) ∈ Fin)
30		xpfi 9311	. . . . . 6 ⊢ ((𝒫 ∪ (M “ 𝑥) ∈ Fin ∧ 𝒫 ∪ (M “ 𝑥) ∈ Fin) → (𝒫 ∪ (M “ 𝑥) × 𝒫 ∪ (M “ 𝑥)) ∈ Fin)
31	29, 29, 30	syl2anc 596	. . . . 5 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 (M‘𝑦) ∈ Fin) → (𝒫 ∪ (M “ 𝑥) × 𝒫 ∪ (M “ 𝑥)) ∈ Fin)
32		imafi 9307	. . . . 5 ⊢ ((Fun |s ∧ (𝒫 ∪ (M “ 𝑥) × 𝒫 ∪ (M “ 𝑥)) ∈ Fin) → ( |s “ (𝒫 ∪ (M “ 𝑥) × 𝒫 ∪ (M “ 𝑥))) ∈ Fin)
33	11, 31, 32	sylancr 599	. . . 4 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 (M‘𝑦) ∈ Fin) → ( |s “ (𝒫 ∪ (M “ 𝑥) × 𝒫 ∪ (M “ 𝑥))) ∈ Fin)
34	8, 33	eqeltrd 2860	. . 3 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 (M‘𝑦) ∈ Fin) → (M‘𝑥) ∈ Fin)
35	34	ex 418	. 2 ⊢ (𝑥 ∈ ω → (∀𝑦 ∈ 𝑥 (M‘𝑦) ∈ Fin → (M‘𝑥) ∈ Fin))
36	2, 4, 35	omsinds 7898	1 ⊢ (𝐴 ∈ ω → (M‘𝐴) ∈ Fin)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895  𝒫 cpw 4556  ∪ cuni 4866   × cxp 5648  dom cdm 5650   “ cima 5653  Oncon0 6362  Fun wfun 6532  ⟶wf 6534  ‘cfv 6538  ωcom 7877  Fincfn 8973  Nocsur 27997   <<s cslts 28143   |s ccuts 28145  Mcmade 28208

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-pow 5326  ax-pr 5390  ax-un 7751

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-rmo 3365  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 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  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-se 5604  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-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-1o 8476  df-2o 8477  df-en 8974  df-dom 8975  df-fin 8977  df-no 28000  df-lts 28001  df-bday 28002  df-slts 28144  df-cuts 28146  df-made 28213

This theorem is used by:  oldfi  28300