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Theorem madeoldsuc 27923
Description: The made set is the old set of its successor. (Contributed by Scott Fenton, 8-Aug-2024.)
Assertion
Ref Expression
madeoldsuc (𝐴 ∈ On → ( M ‘𝐴) = ( O ‘suc 𝐴))

Proof of Theorem madeoldsuc
StepHypRef Expression
1 df-suc 6390 . . . . . . . 8 suc 𝐴 = (𝐴 ∪ {𝐴})
21imaeq2i 6076 . . . . . . 7 ( M “ suc 𝐴) = ( M “ (𝐴 ∪ {𝐴}))
3 imaundi 6169 . . . . . . 7 ( M “ (𝐴 ∪ {𝐴})) = (( M “ 𝐴) ∪ ( M “ {𝐴}))
42, 3eqtri 2765 . . . . . 6 ( M “ suc 𝐴) = (( M “ 𝐴) ∪ ( M “ {𝐴}))
54unieqi 4919 . . . . 5 ( M “ suc 𝐴) = (( M “ 𝐴) ∪ ( M “ {𝐴}))
6 uniun 4930 . . . . 5 (( M “ 𝐴) ∪ ( M “ {𝐴})) = ( ( M “ 𝐴) ∪ ( M “ {𝐴}))
75, 6eqtri 2765 . . . 4 ( M “ suc 𝐴) = ( ( M “ 𝐴) ∪ ( M “ {𝐴}))
87a1i 11 . . 3 (𝐴 ∈ On → ( M “ suc 𝐴) = ( ( M “ 𝐴) ∪ ( M “ {𝐴})))
9 oldval 27893 . . . . 5 (𝐴 ∈ On → ( O ‘𝐴) = ( M “ 𝐴))
109eqcomd 2743 . . . 4 (𝐴 ∈ On → ( M “ 𝐴) = ( O ‘𝐴))
11 madef 27895 . . . . . . . 8 M :On⟶𝒫 No
12 ffn 6736 . . . . . . . 8 ( M :On⟶𝒫 No → M Fn On)
1311, 12ax-mp 5 . . . . . . 7 M Fn On
14 fnsnfv 6988 . . . . . . . 8 (( M Fn On ∧ 𝐴 ∈ On) → {( M ‘𝐴)} = ( M “ {𝐴}))
1514eqcomd 2743 . . . . . . 7 (( M Fn On ∧ 𝐴 ∈ On) → ( M “ {𝐴}) = {( M ‘𝐴)})
1613, 15mpan 690 . . . . . 6 (𝐴 ∈ On → ( M “ {𝐴}) = {( M ‘𝐴)})
1716unieqd 4920 . . . . 5 (𝐴 ∈ On → ( M “ {𝐴}) = {( M ‘𝐴)})
18 fvex 6919 . . . . . 6 ( M ‘𝐴) ∈ V
1918unisn 4926 . . . . 5 {( M ‘𝐴)} = ( M ‘𝐴)
2017, 19eqtrdi 2793 . . . 4 (𝐴 ∈ On → ( M “ {𝐴}) = ( M ‘𝐴))
2110, 20uneq12d 4169 . . 3 (𝐴 ∈ On → ( ( M “ 𝐴) ∪ ( M “ {𝐴})) = (( O ‘𝐴) ∪ ( M ‘𝐴)))
22 oldssmade 27916 . . . . 5 ( O ‘𝐴) ⊆ ( M ‘𝐴)
2322a1i 11 . . . 4 (𝐴 ∈ On → ( O ‘𝐴) ⊆ ( M ‘𝐴))
24 ssequn1 4186 . . . 4 (( O ‘𝐴) ⊆ ( M ‘𝐴) ↔ (( O ‘𝐴) ∪ ( M ‘𝐴)) = ( M ‘𝐴))
2523, 24sylib 218 . . 3 (𝐴 ∈ On → (( O ‘𝐴) ∪ ( M ‘𝐴)) = ( M ‘𝐴))
268, 21, 253eqtrrd 2782 . 2 (𝐴 ∈ On → ( M ‘𝐴) = ( M “ suc 𝐴))
27 onsuc 7831 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ On)
28 oldval 27893 . . 3 (suc 𝐴 ∈ On → ( O ‘suc 𝐴) = ( M “ suc 𝐴))
2927, 28syl 17 . 2 (𝐴 ∈ On → ( O ‘suc 𝐴) = ( M “ suc 𝐴))
3026, 29eqtr4d 2780 1 (𝐴 ∈ On → ( M ‘𝐴) = ( O ‘suc 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cun 3949  wss 3951  𝒫 cpw 4600  {csn 4626   cuni 4907  cima 5688  Oncon0 6384  suc csuc 6386   Fn wfn 6556  wf 6557  cfv 6561   No csur 27684   M cmade 27881   O cold 27882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689  df-sslt 27826  df-scut 27828  df-made 27886  df-old 27887
This theorem is referenced by:  oldsuc  27924  oldlim  27925
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