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Theorem madeoldsuc 27938
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 6392 . . . . . . . 8 suc 𝐴 = (𝐴 ∪ {𝐴})
21imaeq2i 6078 . . . . . . 7 ( M “ suc 𝐴) = ( M “ (𝐴 ∪ {𝐴}))
3 imaundi 6172 . . . . . . 7 ( M “ (𝐴 ∪ {𝐴})) = (( M “ 𝐴) ∪ ( M “ {𝐴}))
42, 3eqtri 2763 . . . . . 6 ( M “ suc 𝐴) = (( M “ 𝐴) ∪ ( M “ {𝐴}))
54unieqi 4924 . . . . 5 ( M “ suc 𝐴) = (( M “ 𝐴) ∪ ( M “ {𝐴}))
6 uniun 4935 . . . . 5 (( M “ 𝐴) ∪ ( M “ {𝐴})) = ( ( M “ 𝐴) ∪ ( M “ {𝐴}))
75, 6eqtri 2763 . . . 4 ( M “ suc 𝐴) = ( ( M “ 𝐴) ∪ ( M “ {𝐴}))
87a1i 11 . . 3 (𝐴 ∈ On → ( M “ suc 𝐴) = ( ( M “ 𝐴) ∪ ( M “ {𝐴})))
9 oldval 27908 . . . . 5 (𝐴 ∈ On → ( O ‘𝐴) = ( M “ 𝐴))
109eqcomd 2741 . . . 4 (𝐴 ∈ On → ( M “ 𝐴) = ( O ‘𝐴))
11 madef 27910 . . . . . . . 8 M :On⟶𝒫 No
12 ffn 6737 . . . . . . . 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 2741 . . . . . . 7 (( M Fn On ∧ 𝐴 ∈ On) → ( M “ {𝐴}) = {( M ‘𝐴)})
1613, 15mpan 690 . . . . . 6 (𝐴 ∈ On → ( M “ {𝐴}) = {( M ‘𝐴)})
1716unieqd 4925 . . . . 5 (𝐴 ∈ On → ( M “ {𝐴}) = {( M ‘𝐴)})
18 fvex 6920 . . . . . 6 ( M ‘𝐴) ∈ V
1918unisn 4931 . . . . 5 {( M ‘𝐴)} = ( M ‘𝐴)
2017, 19eqtrdi 2791 . . . 4 (𝐴 ∈ On → ( M “ {𝐴}) = ( M ‘𝐴))
2110, 20uneq12d 4179 . . 3 (𝐴 ∈ On → ( ( M “ 𝐴) ∪ ( M “ {𝐴})) = (( O ‘𝐴) ∪ ( M ‘𝐴)))
22 oldssmade 27931 . . . . 5 ( O ‘𝐴) ⊆ ( M ‘𝐴)
2322a1i 11 . . . 4 (𝐴 ∈ On → ( O ‘𝐴) ⊆ ( M ‘𝐴))
24 ssequn1 4196 . . . 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 2780 . 2 (𝐴 ∈ On → ( M ‘𝐴) = ( M “ suc 𝐴))
27 onsuc 7831 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ On)
28 oldval 27908 . . 3 (suc 𝐴 ∈ On → ( O ‘suc 𝐴) = ( M “ suc 𝐴))
2927, 28syl 17 . 2 (𝐴 ∈ On → ( O ‘suc 𝐴) = ( M “ suc 𝐴))
3026, 29eqtr4d 2778 1 (𝐴 ∈ On → ( M ‘𝐴) = ( O ‘suc 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cun 3961  wss 3963  𝒫 cpw 4605  {csn 4631   cuni 4912  cima 5692  Oncon0 6386  suc csuc 6388   Fn wfn 6558  wf 6559  cfv 6563   No csur 27699   M cmade 27896   O cold 27897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sslt 27841  df-scut 27843  df-made 27901  df-old 27902
This theorem is referenced by:  oldsuc  27939  oldlim  27940
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