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Theorem madef 27896
Description: The made function is a function from ordinals to sets of surreals. (Contributed by Scott Fenton, 6-Aug-2024.)
Assertion
Ref Expression
madef M :On⟶𝒫 No

Proof of Theorem madef
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-made 27887 . . 3 M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥))))
21tfr1 8438 . 2 M Fn On
3 madeval2 27893 . . . . . . 7 (𝑥 ∈ On → ( M ‘𝑥) = {𝑦 No ∣ ∃𝑧 ∈ 𝒫 ( M “ 𝑥)∃𝑤 ∈ 𝒫 ( M “ 𝑥)(𝑧 <<s 𝑤 ∧ (𝑧 |s 𝑤) = 𝑦)})
4 ssrab2 4079 . . . . . . 7 {𝑦 No ∣ ∃𝑧 ∈ 𝒫 ( M “ 𝑥)∃𝑤 ∈ 𝒫 ( M “ 𝑥)(𝑧 <<s 𝑤 ∧ (𝑧 |s 𝑤) = 𝑦)} ⊆ No
53, 4eqsstrdi 4027 . . . . . 6 (𝑥 ∈ On → ( M ‘𝑥) ⊆ No )
6 sseq1 4008 . . . . . 6 (𝑦 = ( M ‘𝑥) → (𝑦 No ↔ ( M ‘𝑥) ⊆ No ))
75, 6syl5ibrcom 247 . . . . 5 (𝑥 ∈ On → (𝑦 = ( M ‘𝑥) → 𝑦 No ))
87rexlimiv 3147 . . . 4 (∃𝑥 ∈ On 𝑦 = ( M ‘𝑥) → 𝑦 No )
9 vex 3483 . . . . 5 𝑦 ∈ V
10 eqeq1 2740 . . . . . 6 (𝑧 = 𝑦 → (𝑧 = ( M ‘𝑥) ↔ 𝑦 = ( M ‘𝑥)))
1110rexbidv 3178 . . . . 5 (𝑧 = 𝑦 → (∃𝑥 ∈ On 𝑧 = ( M ‘𝑥) ↔ ∃𝑥 ∈ On 𝑦 = ( M ‘𝑥)))
12 fnrnfv 6967 . . . . . 6 ( M Fn On → ran M = {𝑧 ∣ ∃𝑥 ∈ On 𝑧 = ( M ‘𝑥)})
132, 12ax-mp 5 . . . . 5 ran M = {𝑧 ∣ ∃𝑥 ∈ On 𝑧 = ( M ‘𝑥)}
149, 11, 13elab2 3681 . . . 4 (𝑦 ∈ ran M ↔ ∃𝑥 ∈ On 𝑦 = ( M ‘𝑥))
15 velpw 4604 . . . 4 (𝑦 ∈ 𝒫 No 𝑦 No )
168, 14, 153imtr4i 292 . . 3 (𝑦 ∈ ran M → 𝑦 ∈ 𝒫 No )
1716ssriv 3986 . 2 ran M ⊆ 𝒫 No
18 df-f 6564 . 2 ( M :On⟶𝒫 No ↔ ( M Fn On ∧ ran M ⊆ 𝒫 No ))
192, 17, 18mpbir2an 711 1 M :On⟶𝒫 No
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1539  wcel 2107  {cab 2713  wrex 3069  {crab 3435  Vcvv 3479  wss 3950  𝒫 cpw 4599   cuni 4906   class class class wbr 5142  cmpt 5224   × cxp 5682  ran crn 5685  cima 5687  Oncon0 6383   Fn wfn 6555  wf 6556  cfv 6560  (class class class)co 7432   No csur 27685   <<s csslt 27826   |s cscut 27828   M cmade 27882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-1o 8507  df-2o 8508  df-no 27688  df-slt 27689  df-bday 27690  df-sslt 27827  df-scut 27829  df-made 27887
This theorem is referenced by:  oldf  27897  newf  27898  madessno  27900  elmade  27907  elold  27909  old1  27915  madess  27916  madeoldsuc  27924  madebdayim  27927  madefi  27951  oldfi  27952
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