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Theorem oldf 33804
Description: The older function is a function from ordinals to sets of surreals. (Contributed by Scott Fenton, 6-Aug-2024.)
Assertion
Ref Expression
oldf O :On⟶𝒫 No

Proof of Theorem oldf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-old 33795 . 2 O = (𝑥 ∈ On ↦ ( M “ 𝑥))
2 imassrn 5955 . . . . . . . 8 ( M “ 𝑥) ⊆ ran M
3 madef 33803 . . . . . . . . 9 M :On⟶𝒫 No
4 frn 6571 . . . . . . . . 9 ( M :On⟶𝒫 No → ran M ⊆ 𝒫 No )
53, 4ax-mp 5 . . . . . . . 8 ran M ⊆ 𝒫 No
62, 5sstri 3925 . . . . . . 7 ( M “ 𝑥) ⊆ 𝒫 No
76sseli 3911 . . . . . 6 (𝑦 ∈ ( M “ 𝑥) → 𝑦 ∈ 𝒫 No )
87elpwid 4539 . . . . 5 (𝑦 ∈ ( M “ 𝑥) → 𝑦 No )
98rgen 3072 . . . 4 𝑦 ∈ ( M “ 𝑥)𝑦 No
109a1i 11 . . 3 (𝑥 ∈ On → ∀𝑦 ∈ ( M “ 𝑥)𝑦 No )
11 ffun 6567 . . . . . . . 8 ( M :On⟶𝒫 No → Fun M )
123, 11ax-mp 5 . . . . . . 7 Fun M
13 vex 3425 . . . . . . . 8 𝑥 ∈ V
1413funimaex 6485 . . . . . . 7 (Fun M → ( M “ 𝑥) ∈ V)
1512, 14ax-mp 5 . . . . . 6 ( M “ 𝑥) ∈ V
1615uniex 7548 . . . . 5 ( M “ 𝑥) ∈ V
1716elpw 4532 . . . 4 ( ( M “ 𝑥) ∈ 𝒫 No ( M “ 𝑥) ⊆ No )
18 unissb 4868 . . . 4 ( ( M “ 𝑥) ⊆ No ↔ ∀𝑦 ∈ ( M “ 𝑥)𝑦 No )
1917, 18bitri 278 . . 3 ( ( M “ 𝑥) ∈ 𝒫 No ↔ ∀𝑦 ∈ ( M “ 𝑥)𝑦 No )
2010, 19sylibr 237 . 2 (𝑥 ∈ On → ( M “ 𝑥) ∈ 𝒫 No )
211, 20fmpti 6948 1 O :On⟶𝒫 No
Colors of variables: wff setvar class
Syntax hints:  wcel 2111  wral 3062  Vcvv 3421  wss 3881  𝒫 cpw 4528   cuni 4834  ran crn 5567  cima 5569  Oncon0 6231  Fun wfun 6392  wf 6394   No csur 33606   M cmade 33789   O cold 33790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-rep 5194  ax-sep 5207  ax-nul 5214  ax-pow 5273  ax-pr 5337  ax-un 7542
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-reu 3069  df-rmo 3070  df-rab 3071  df-v 3423  df-sbc 3710  df-csb 3827  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4253  df-if 4455  df-pw 4530  df-sn 4557  df-pr 4559  df-tp 4561  df-op 4563  df-uni 4835  df-int 4875  df-iun 4921  df-br 5069  df-opab 5131  df-mpt 5151  df-tr 5177  df-id 5470  df-eprel 5475  df-po 5483  df-so 5484  df-fr 5524  df-we 5526  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-rn 5577  df-res 5578  df-ima 5579  df-pred 6176  df-ord 6234  df-on 6235  df-suc 6237  df-iota 6356  df-fun 6400  df-fn 6401  df-f 6402  df-f1 6403  df-fo 6404  df-f1o 6405  df-fv 6406  df-riota 7189  df-ov 7235  df-oprab 7236  df-mpo 7237  df-wrecs 8068  df-recs 8129  df-1o 8223  df-2o 8224  df-no 33609  df-slt 33610  df-bday 33611  df-sslt 33739  df-scut 33741  df-made 33794  df-old 33795
This theorem is referenced by:  oldssno  33808  leftf  33812  rightf  33813  oldssmade  33823  oldlim  33832  oldbdayim  33834
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