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 6829	. . 3 ⊢ (𝑥 = 𝑦 → ( M ‘𝑥) = ( M ‘𝑦))
2	1	eleq1d 2820	. 2 ⊢ (𝑥 = 𝑦 → (( M ‘𝑥) ∈ Fin ↔ ( M ‘𝑦) ∈ Fin))
3		fveq2 6829	. . 3 ⊢ (𝑥 = 𝐴 → ( M ‘𝑥) = ( M ‘𝐴))
4	3	eleq1d 2820	. 2 ⊢ (𝑥 = 𝐴 → (( M ‘𝑥) ∈ Fin ↔ ( M ‘𝐴) ∈ Fin))
5		nnon 7812	. . . . . 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 6660	. . . . . . . . . . 11 ⊢ ( M :On⟶𝒫 No → Fun M )
11	9, 10	ax-mp 5	. . . . . . . . . 10 ⊢ Fun M
12		nnfi 9091	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → 𝑥 ∈ Fin)
13		imafi 9214	. . . . . . . . . 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 7728	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
17	5, 16	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ω → 𝑥 ⊆ On)
18	9	fdmi 6668	. . . . . . . . . . 11 ⊢ dom M = On
19	17, 18	sseqtrrdi 3958	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → 𝑥 ⊆ dom M )
20		funimass4 6893	. . . . . . . . . 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 9243	. . . . . . . 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 9218	. . . . . . 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 9219	. . . . . 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 3431	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
30	29	funimaex 6575	. . . . . . . . . 10 ⊢ (Fun M → ( M “ 𝑥) ∈ V)
31	11, 30	ax-mp 5	. . . . . . . . 9 ⊢ ( M “ 𝑥) ∈ V
32	31	uniex 7684	. . . . . . . 8 ⊢ ∪ ( M “ 𝑥) ∈ V
33	32	pwex 5311	. . . . . . 7 ⊢ 𝒫 ∪ ( M “ 𝑥) ∈ V
34	33, 33	xpex 7696	. . . . . 6 ⊢ (𝒫 ∪ ( M “ 𝑥) × 𝒫 ∪ ( M “ 𝑥)) ∈ V
35		cutsf 27772	. . . . . . 7 ⊢ |s : <<s ⟶ No
36		ffun 6660	. . . . . . 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 9112	. . . . 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 2835	. . 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 7827	1 ⊢ (𝐴 ∈ ω → ( M ‘𝐴) ∈ Fin)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3049  Vcvv 3427   ⊆ wss 3885  𝒫 cpw 4531  ∪ cuni 4840   class class class wbr 5074   × cxp 5618  dom cdm 5620   “ cima 5623  Oncon0 6312  Fun wfun 6481  ⟶wf 6483  ‘cfv 6487  ωcom 7806   ≼ cdom 8880  Fincfn 8882   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 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678  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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rmo 3340  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-se 5574  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-isom 6496  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-1o 8394  df-2o 8395  df-er 8632  df-map 8764  df-en 8883  df-dom 8884  df-fin 8886  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