Theorem madefi 27968

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 6920	. . 3 ⊢ (𝑥 = 𝑦 → ( M ‘𝑥) = ( M ‘𝑦))
2	1	eleq1d 2829	. 2 ⊢ (𝑥 = 𝑦 → (( M ‘𝑥) ∈ Fin ↔ ( M ‘𝑦) ∈ Fin))
3		fveq2 6920	. . 3 ⊢ (𝑥 = 𝐴 → ( M ‘𝑥) = ( M ‘𝐴))
4	3	eleq1d 2829	. 2 ⊢ (𝑥 = 𝐴 → (( M ‘𝑥) ∈ Fin ↔ ( M ‘𝐴) ∈ Fin))
5		nnon 7909	. . . . . 6 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On)
6		madeval 27909	. . . . . 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 27913	. . . . . . . . . . 11 ⊢ M :On⟶𝒫 No
10		ffun 6750	. . . . . . . . . . 11 ⊢ ( M :On⟶𝒫 No → Fun M )
11	9, 10	ax-mp 5	. . . . . . . . . 10 ⊢ Fun M
12		nnfi 9233	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → 𝑥 ∈ Fin)
13		imafi 9381	. . . . . . . . . 10 ⊢ ((Fun M ∧ 𝑥 ∈ Fin) → ( M “ 𝑥) ∈ Fin)
14	11, 12, 13	sylancr 586	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → ( M “ 𝑥) ∈ Fin)
15	14	adantr 480	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin) → ( M “ 𝑥) ∈ Fin)
16		onss 7820	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
17	5, 16	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ω → 𝑥 ⊆ On)
18	9	fdmi 6758	. . . . . . . . . . 11 ⊢ dom M = On
19	17, 18	sseqtrrdi 4060	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → 𝑥 ⊆ dom M )
20		funimass4 6986	. . . . . . . . . 10 ⊢ ((Fun M ∧ 𝑥 ⊆ dom M ) → (( M “ 𝑥) ⊆ Fin ↔ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin))
21	11, 19, 20	sylancr 586	. . . . . . . . 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 9412	. . . . . . . 8 ⊢ ((( M “ 𝑥) ∈ Fin ∧ ( M “ 𝑥) ⊆ Fin) → ∪ ( M “ 𝑥) ∈ Fin)
24	15, 22, 23	syl2anc 583	. . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin) → ∪ ( M “ 𝑥) ∈ Fin)
25		pwfi 9385	. . . . . . 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 9386	. . . . . 6 ⊢ ((𝒫 ∪ ( M “ 𝑥) ∈ Fin ∧ 𝒫 ∪ ( M “ 𝑥) ∈ Fin) → (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥)) ∈ Fin)
28	26, 26, 27	syl2anc 583	. . . . 5 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin) → (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥)) ∈ Fin)
29		vex 3492	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
30	29	funimaex 6666	. . . . . . . . . 10 ⊢ (Fun M → ( M “ 𝑥) ∈ V)
31	11, 30	ax-mp 5	. . . . . . . . 9 ⊢ ( M “ 𝑥) ∈ V
32	31	uniex 7776	. . . . . . . 8 ⊢ ∪ ( M “ 𝑥) ∈ V
33	32	pwex 5398	. . . . . . 7 ⊢ 𝒫 ∪ ( M “ 𝑥) ∈ V
34	33, 33	xpex 7788	. . . . . 6 ⊢ (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥)) ∈ V
35		scutf 27875	. . . . . . 7 ⊢ |s : <<s ⟶ No
36		ffun 6750	. . . . . . 7 ⊢ ( |s : <<s ⟶ No → Fun |s )
37	35, 36	ax-mp 5	. . . . . 6 ⊢ Fun |s
38		imadomg 10603	. . . . . 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 9255	. . . . 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 585	. . . 4 ⊢ ((𝑥 ∈ ω ∧ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin) → ( |s “ (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥))) ∈ Fin)
42	8, 41	eqeltrd 2844	. . 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 7924	1 ⊢ (𝐴 ∈ ω → ( M ‘𝐴) ∈ Fin)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ⊆ wss 3976  𝒫 cpw 4622  ∪ cuni 4931   class class class wbr 5166   × cxp 5698  dom cdm 5700   “ cima 5703  Oncon0 6395  Fun wfun 6567  ⟶wf 6569  ‘cfv 6573  ωcom 7903   ≼ cdom 9001  Fincfn 9003   No csur 27702   <<s csslt 27843   |s cscut 27845   M cmade 27899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-ac2 10532

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-fin 9007  df-card 10008  df-acn 10011  df-ac 10185  df-no 27705  df-slt 27706  df-bday 27707  df-sslt 27844  df-scut 27846  df-made 27904

This theorem is referenced by:  oldfi  27969