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Theorem madef 27910
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 27901 . . 3 M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥))))
21tfr1 8436 . 2 M Fn On
3 madeval2 27907 . . . . . . 7 (𝑥 ∈ On → ( M ‘𝑥) = {𝑦 No ∣ ∃𝑧 ∈ 𝒫 ( M “ 𝑥)∃𝑤 ∈ 𝒫 ( M “ 𝑥)(𝑧 <<s 𝑤 ∧ (𝑧 |s 𝑤) = 𝑦)})
4 ssrab2 4090 . . . . . . 7 {𝑦 No ∣ ∃𝑧 ∈ 𝒫 ( M “ 𝑥)∃𝑤 ∈ 𝒫 ( M “ 𝑥)(𝑧 <<s 𝑤 ∧ (𝑧 |s 𝑤) = 𝑦)} ⊆ No
53, 4eqsstrdi 4050 . . . . . 6 (𝑥 ∈ On → ( M ‘𝑥) ⊆ No )
6 sseq1 4021 . . . . . 6 (𝑦 = ( M ‘𝑥) → (𝑦 No ↔ ( M ‘𝑥) ⊆ No ))
75, 6syl5ibrcom 247 . . . . 5 (𝑥 ∈ On → (𝑦 = ( M ‘𝑥) → 𝑦 No ))
87rexlimiv 3146 . . . 4 (∃𝑥 ∈ On 𝑦 = ( M ‘𝑥) → 𝑦 No )
9 vex 3482 . . . . 5 𝑦 ∈ V
10 eqeq1 2739 . . . . . 6 (𝑧 = 𝑦 → (𝑧 = ( M ‘𝑥) ↔ 𝑦 = ( M ‘𝑥)))
1110rexbidv 3177 . . . . 5 (𝑧 = 𝑦 → (∃𝑥 ∈ On 𝑧 = ( M ‘𝑥) ↔ ∃𝑥 ∈ On 𝑦 = ( M ‘𝑥)))
12 fnrnfv 6968 . . . . . 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 3685 . . . 4 (𝑦 ∈ ran M ↔ ∃𝑥 ∈ On 𝑦 = ( M ‘𝑥))
15 velpw 4610 . . . 4 (𝑦 ∈ 𝒫 No 𝑦 No )
168, 14, 153imtr4i 292 . . 3 (𝑦 ∈ ran M → 𝑦 ∈ 𝒫 No )
1716ssriv 3999 . 2 ran M ⊆ 𝒫 No
18 df-f 6567 . 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 1537  wcel 2106  {cab 2712  wrex 3068  {crab 3433  Vcvv 3478  wss 3963  𝒫 cpw 4605   cuni 4912   class class class wbr 5148  cmpt 5231   × cxp 5687  ran crn 5690  cima 5692  Oncon0 6386   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431   No csur 27699   <<s csslt 27840   |s cscut 27842   M cmade 27896
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
This theorem is referenced by:  oldf  27911  newf  27912  madessno  27914  elmade  27921  elold  27923  old1  27929  madess  27930  madeoldsuc  27938  madebdayim  27941  madefi  27965  oldfi  27966
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