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Theorem madef 34136
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 :OnβŸΆπ’« No

Proof of Theorem madef
Dummy variables π‘₯ 𝑦 𝑧 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-made 34127 . . 3 M = recs((π‘₯ ∈ V ↦ ( |s β€œ (𝒫 βˆͺ ran π‘₯ Γ— 𝒫 βˆͺ ran π‘₯))))
21tfr1 8298 . 2 M Fn On
3 madeval2 34133 . . . . . . 7 (π‘₯ ∈ On β†’ ( M β€˜π‘₯) = {𝑦 ∈ No ∣ βˆƒπ‘§ ∈ 𝒫 βˆͺ ( M β€œ π‘₯)βˆƒπ‘€ ∈ 𝒫 βˆͺ ( M β€œ π‘₯)(𝑧 <<s 𝑀 ∧ (𝑧 |s 𝑀) = 𝑦)})
4 ssrab2 4025 . . . . . . 7 {𝑦 ∈ No ∣ βˆƒπ‘§ ∈ 𝒫 βˆͺ ( M β€œ π‘₯)βˆƒπ‘€ ∈ 𝒫 βˆͺ ( M β€œ π‘₯)(𝑧 <<s 𝑀 ∧ (𝑧 |s 𝑀) = 𝑦)} βŠ† No
53, 4eqsstrdi 3986 . . . . . 6 (π‘₯ ∈ On β†’ ( M β€˜π‘₯) βŠ† No )
6 sseq1 3957 . . . . . 6 (𝑦 = ( M β€˜π‘₯) β†’ (𝑦 βŠ† No ↔ ( M β€˜π‘₯) βŠ† No ))
75, 6syl5ibrcom 246 . . . . 5 (π‘₯ ∈ On β†’ (𝑦 = ( M β€˜π‘₯) β†’ 𝑦 βŠ† No ))
87rexlimiv 3141 . . . 4 (βˆƒπ‘₯ ∈ On 𝑦 = ( M β€˜π‘₯) β†’ 𝑦 βŠ† No )
9 vex 3445 . . . . 5 𝑦 ∈ V
10 eqeq1 2740 . . . . . 6 (𝑧 = 𝑦 β†’ (𝑧 = ( M β€˜π‘₯) ↔ 𝑦 = ( M β€˜π‘₯)))
1110rexbidv 3171 . . . . 5 (𝑧 = 𝑦 β†’ (βˆƒπ‘₯ ∈ On 𝑧 = ( M β€˜π‘₯) ↔ βˆƒπ‘₯ ∈ On 𝑦 = ( M β€˜π‘₯)))
12 fnrnfv 6885 . . . . . 6 ( M Fn On β†’ ran M = {𝑧 ∣ βˆƒπ‘₯ ∈ On 𝑧 = ( M β€˜π‘₯)})
132, 12ax-mp 5 . . . . 5 ran M = {𝑧 ∣ βˆƒπ‘₯ ∈ On 𝑧 = ( M β€˜π‘₯)}
149, 11, 13elab2 3623 . . . 4 (𝑦 ∈ ran M ↔ βˆƒπ‘₯ ∈ On 𝑦 = ( M β€˜π‘₯))
15 velpw 4552 . . . 4 (𝑦 ∈ 𝒫 No ↔ 𝑦 βŠ† No )
168, 14, 153imtr4i 291 . . 3 (𝑦 ∈ ran M β†’ 𝑦 ∈ 𝒫 No )
1716ssriv 3936 . 2 ran M βŠ† 𝒫 No
18 df-f 6483 . 2 ( M :OnβŸΆπ’« No ↔ ( M Fn On ∧ ran M βŠ† 𝒫 No ))
192, 17, 18mpbir2an 708 1 M :OnβŸΆπ’« No
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 396   = wceq 1540   ∈ wcel 2105  {cab 2713  βˆƒwrex 3070  {crab 3403  Vcvv 3441   βŠ† wss 3898  π’« cpw 4547  βˆͺ cuni 4852   class class class wbr 5092   ↦ cmpt 5175   Γ— cxp 5618  ran crn 5621   β€œ cima 5623  Oncon0 6302   Fn wfn 6474  βŸΆwf 6475  β€˜cfv 6479  (class class class)co 7337   No csur 26894   <<s csslt 27026   |s cscut 27028   M cmade 34122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4853  df-int 4895  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-1o 8367  df-2o 8368  df-no 26897  df-slt 26898  df-bday 26899  df-sslt 27027  df-scut 27029  df-made 34127
This theorem is referenced by:  oldf  34137  newf  34138  madessno  34140  elmade  34147  elold  34149  madess  34155  madeoldsuc  34163  madebdayim  34166
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