Mathbox for Scott Fenton < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  madeoldsuc Structured version   Visualization version   GIF version

 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))

StepHypRef Expression
1 df-suc 6175 . . . . . . . 8 suc 𝐴 = (𝐴 ∪ {𝐴})
21imaeq2i 5899 . . . . . . 7 ( M “ suc 𝐴) = ( M “ (𝐴 ∪ {𝐴}))
3 imaundi 5980 . . . . . . 7 ( M “ (𝐴 ∪ {𝐴})) = (( M “ 𝐴) ∪ ( M “ {𝐴}))
42, 3eqtri 2781 . . . . . 6 ( M “ suc 𝐴) = (( M “ 𝐴) ∪ ( M “ {𝐴}))
54unieqi 4811 . . . . 5 ( M “ suc 𝐴) = (( M “ 𝐴) ∪ ( M “ {𝐴}))
6 uniun 4823 . . . . 5 (( M “ 𝐴) ∪ ( M “ {𝐴})) = ( ( M “ 𝐴) ∪ ( M “ {𝐴}))
75, 6eqtri 2781 . . . 4 ( M “ suc 𝐴) = ( ( M “ 𝐴) ∪ ( M “ {𝐴}))
87a1i 11 . . 3 (𝐴 ∈ On → ( M “ suc 𝐴) = ( ( M “ 𝐴) ∪ ( M “ {𝐴})))
9 oldval 33598 . . . . 5 (𝐴 ∈ On → ( O ‘𝐴) = ( M “ 𝐴))
109eqcomd 2764 . . . 4 (𝐴 ∈ On → ( M “ 𝐴) = ( O ‘𝐴))
11 madef 33600 . . . . . . . 8 M :On⟶𝒫 No
12 ffn 6498 . . . . . . . 8 ( M :On⟶𝒫 No → M Fn On)
1311, 12ax-mp 5 . . . . . . 7 M Fn On
14 fnsnfv 6731 . . . . . . . 8 (( M Fn On ∧ 𝐴 ∈ On) → {( M ‘𝐴)} = ( M “ {𝐴}))
1514eqcomd 2764 . . . . . . 7 (( M Fn On ∧ 𝐴 ∈ On) → ( M “ {𝐴}) = {( M ‘𝐴)})
1613, 15mpan 689 . . . . . 6 (𝐴 ∈ On → ( M “ {𝐴}) = {( M ‘𝐴)})
1716unieqd 4812 . . . . 5 (𝐴 ∈ On → ( M “ {𝐴}) = {( M ‘𝐴)})
18 fvex 6671 . . . . . 6 ( M ‘𝐴) ∈ V
1918unisn 4820 . . . . 5 {( M ‘𝐴)} = ( M ‘𝐴)
2017, 19eqtrdi 2809 . . . 4 (𝐴 ∈ On → ( M “ {𝐴}) = ( M ‘𝐴))
2110, 20uneq12d 4069 . . 3 (𝐴 ∈ On → ( ( M “ 𝐴) ∪ ( M “ {𝐴})) = (( O ‘𝐴) ∪ ( M ‘𝐴)))
22 oldssmade 33617 . . . 4 (𝐴 ∈ On → ( O ‘𝐴) ⊆ ( M ‘𝐴))
23 ssequn1 4085 . . . 4 (( O ‘𝐴) ⊆ ( M ‘𝐴) ↔ (( O ‘𝐴) ∪ ( M ‘𝐴)) = ( M ‘𝐴))
2422, 23sylib 221 . . 3 (𝐴 ∈ On → (( O ‘𝐴) ∪ ( M ‘𝐴)) = ( M ‘𝐴))
258, 21, 243eqtrrd 2798 . 2 (𝐴 ∈ On → ( M ‘𝐴) = ( M “ suc 𝐴))
26 suceloni 7527 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ On)
27 oldval 33598 . . 3 (suc 𝐴 ∈ On → ( O ‘suc 𝐴) = ( M “ suc 𝐴))
2826, 27syl 17 . 2 (𝐴 ∈ On → ( O ‘suc 𝐴) = ( M “ suc 𝐴))
2925, 28eqtr4d 2796 1 (𝐴 ∈ On → ( M ‘𝐴) = ( O ‘suc 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∪ cun 3856   ⊆ wss 3858  𝒫 cpw 4494  {csn 4522  ∪ cuni 4798   “ cima 5527  Oncon0 6169  suc csuc 6171   Fn wfn 6330  ⟶wf 6331  ‘cfv 6335   No csur 33408   M cmade 33586   O cold 33587 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-wrecs 7957  df-recs 8018  df-1o 8112  df-2o 8113  df-no 33411  df-slt 33412  df-bday 33413  df-sslt 33541  df-scut 33543  df-made 33591  df-old 33592 This theorem is referenced by:  oldsuc  33625  oldlim  33626
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