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Theorem madefi 27959
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.
StepHypRef Expression
1 fveq2 6919 . . 3 (𝑥 = 𝑦 → ( M ‘𝑥) = ( M ‘𝑦))
21eleq1d 2823 . 2 (𝑥 = 𝑦 → (( M ‘𝑥) ∈ Fin ↔ ( M ‘𝑦) ∈ Fin))
3 fveq2 6919 . . 3 (𝑥 = 𝐴 → ( M ‘𝑥) = ( M ‘𝐴))
43eleq1d 2823 . 2 (𝑥 = 𝐴 → (( M ‘𝑥) ∈ Fin ↔ ( M ‘𝐴) ∈ Fin))
5 nnon 7905 . . . . . 6 (𝑥 ∈ ω → 𝑥 ∈ On)
6 madeval 27900 . . . . . 6 (𝑥 ∈ On → ( M ‘𝑥) = ( |s “ (𝒫 ( M “ 𝑥) × 𝒫 ( M “ 𝑥))))
75, 6syl 17 . . . . 5 (𝑥 ∈ ω → ( M ‘𝑥) = ( |s “ (𝒫 ( M “ 𝑥) × 𝒫 ( M “ 𝑥))))
87adantr 480 . . . 4 ((𝑥 ∈ ω ∧ ∀𝑦𝑥 ( M ‘𝑦) ∈ Fin) → ( M ‘𝑥) = ( |s “ (𝒫 ( M “ 𝑥) × 𝒫 ( M “ 𝑥))))
9 madef 27904 . . . . . . . . . . 11 M :On⟶𝒫 No
10 ffun 6749 . . . . . . . . . . 11 ( M :On⟶𝒫 No → Fun M )
119, 10ax-mp 5 . . . . . . . . . 10 Fun M
12 nnfi 9229 . . . . . . . . . 10 (𝑥 ∈ ω → 𝑥 ∈ Fin)
13 imafi 9377 . . . . . . . . . 10 ((Fun M ∧ 𝑥 ∈ Fin) → ( M “ 𝑥) ∈ Fin)
1411, 12, 13sylancr 586 . . . . . . . . 9 (𝑥 ∈ ω → ( M “ 𝑥) ∈ Fin)
1514adantr 480 . . . . . . . 8 ((𝑥 ∈ ω ∧ ∀𝑦𝑥 ( M ‘𝑦) ∈ Fin) → ( M “ 𝑥) ∈ Fin)
16 onss 7816 . . . . . . . . . . . 12 (𝑥 ∈ On → 𝑥 ⊆ On)
175, 16syl 17 . . . . . . . . . . 11 (𝑥 ∈ ω → 𝑥 ⊆ On)
189fdmi 6757 . . . . . . . . . . 11 dom M = On
1917, 18sseqtrrdi 4054 . . . . . . . . . 10 (𝑥 ∈ ω → 𝑥 ⊆ dom M )
20 funimass4 6985 . . . . . . . . . 10 ((Fun M ∧ 𝑥 ⊆ dom M ) → (( M “ 𝑥) ⊆ Fin ↔ ∀𝑦𝑥 ( M ‘𝑦) ∈ Fin))
2111, 19, 20sylancr 586 . . . . . . . . 9 (𝑥 ∈ ω → (( M “ 𝑥) ⊆ Fin ↔ ∀𝑦𝑥 ( M ‘𝑦) ∈ Fin))
2221biimpar 477 . . . . . . . 8 ((𝑥 ∈ ω ∧ ∀𝑦𝑥 ( M ‘𝑦) ∈ Fin) → ( M “ 𝑥) ⊆ Fin)
23 unifi 9408 . . . . . . . 8 ((( M “ 𝑥) ∈ Fin ∧ ( M “ 𝑥) ⊆ Fin) → ( M “ 𝑥) ∈ Fin)
2415, 22, 23syl2anc 583 . . . . . . 7 ((𝑥 ∈ ω ∧ ∀𝑦𝑥 ( M ‘𝑦) ∈ Fin) → ( M “ 𝑥) ∈ Fin)
25 pwfi 9381 . . . . . . 7 ( ( M “ 𝑥) ∈ Fin ↔ 𝒫 ( M “ 𝑥) ∈ Fin)
2624, 25sylib 218 . . . . . 6 ((𝑥 ∈ ω ∧ ∀𝑦𝑥 ( M ‘𝑦) ∈ Fin) → 𝒫 ( M “ 𝑥) ∈ Fin)
27 xpfi 9382 . . . . . 6 ((𝒫 ( M “ 𝑥) ∈ Fin ∧ 𝒫 ( M “ 𝑥) ∈ Fin) → (𝒫 ( M “ 𝑥) × 𝒫 ( M “ 𝑥)) ∈ Fin)
2826, 26, 27syl2anc 583 . . . . 5 ((𝑥 ∈ ω ∧ ∀𝑦𝑥 ( M ‘𝑦) ∈ Fin) → (𝒫 ( M “ 𝑥) × 𝒫 ( M “ 𝑥)) ∈ Fin)
29 vex 3486 . . . . . . . . . . 11 𝑥 ∈ V
3029funimaex 6665 . . . . . . . . . 10 (Fun M → ( M “ 𝑥) ∈ V)
3111, 30ax-mp 5 . . . . . . . . 9 ( M “ 𝑥) ∈ V
3231uniex 7772 . . . . . . . 8 ( M “ 𝑥) ∈ V
3332pwex 5401 . . . . . . 7 𝒫 ( M “ 𝑥) ∈ V
3433, 33xpex 7784 . . . . . 6 (𝒫 ( M “ 𝑥) × 𝒫 ( M “ 𝑥)) ∈ V
35 scutf 27866 . . . . . . 7 |s : <<s ⟶ No
36 ffun 6749 . . . . . . 7 ( |s : <<s ⟶ No → Fun |s )
3735, 36ax-mp 5 . . . . . 6 Fun |s
38 imadomg 10599 . . . . . 6 ((𝒫 ( M “ 𝑥) × 𝒫 ( M “ 𝑥)) ∈ V → (Fun |s → ( |s “ (𝒫 ( M “ 𝑥) × 𝒫 ( M “ 𝑥))) ≼ (𝒫 ( M “ 𝑥) × 𝒫 ( M “ 𝑥))))
3934, 37, 38mp2 9 . . . . 5 ( |s “ (𝒫 ( M “ 𝑥) × 𝒫 ( M “ 𝑥))) ≼ (𝒫 ( M “ 𝑥) × 𝒫 ( M “ 𝑥))
40 domfi 9251 . . . . 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)
4128, 39, 40sylancl 585 . . . 4 ((𝑥 ∈ ω ∧ ∀𝑦𝑥 ( M ‘𝑦) ∈ Fin) → ( |s “ (𝒫 ( M “ 𝑥) × 𝒫 ( M “ 𝑥))) ∈ Fin)
428, 41eqeltrd 2838 . . 3 ((𝑥 ∈ ω ∧ ∀𝑦𝑥 ( M ‘𝑦) ∈ Fin) → ( M ‘𝑥) ∈ Fin)
4342ex 412 . 2 (𝑥 ∈ ω → (∀𝑦𝑥 ( M ‘𝑦) ∈ Fin → ( M ‘𝑥) ∈ Fin))
442, 4, 43omsinds 7920 1 (𝐴 ∈ ω → ( M ‘𝐴) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2103  wral 3063  Vcvv 3482  wss 3970  𝒫 cpw 4622   cuni 4931   class class class wbr 5169   × cxp 5697  dom cdm 5699  cima 5702  Oncon0 6394  Fun wfun 6566  wf 6568  cfv 6572  ωcom 7899  cdom 8997  Fincfn 8999   No csur 27693   <<s csslt 27834   |s cscut 27836   M cmade 27890
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 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766  ax-ac2 10528
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 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3383  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4973  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-se 5655  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-isom 6581  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-om 7900  df-1st 8026  df-2nd 8027  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-1o 8518  df-2o 8519  df-er 8759  df-map 8882  df-en 9000  df-dom 9001  df-fin 9003  df-card 10004  df-acn 10007  df-ac 10181  df-no 27696  df-slt 27697  df-bday 27698  df-sslt 27835  df-scut 27837  df-made 27895
This theorem is referenced by:  oldfi  27960
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