Theorem madef 27340

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.

Step	Hyp	Ref	Expression
1		df-made 27331	. . 3 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))))
2	1	tfr1 8393	. 2 ⊢ M Fn On
3		madeval2 27337	. . . . . . 7 ⊢ (𝑥 ∈ On → ( M ‘𝑥) = {𝑦 ∈ No ∣ ∃𝑧 ∈ 𝒫 ∪ ( M “ 𝑥)∃𝑤 ∈ 𝒫 ∪ ( M “ 𝑥)(𝑧 <<s 𝑤 ∧ (𝑧 |s 𝑤) = 𝑦)})
4		ssrab2 4076	. . . . . . 7 ⊢ {𝑦 ∈ No ∣ ∃𝑧 ∈ 𝒫 ∪ ( M “ 𝑥)∃𝑤 ∈ 𝒫 ∪ ( M “ 𝑥)(𝑧 <<s 𝑤 ∧ (𝑧 |s 𝑤) = 𝑦)} ⊆ No
5	3, 4	eqsstrdi 4035	. . . . . 6 ⊢ (𝑥 ∈ On → ( M ‘𝑥) ⊆ No )
6		sseq1 4006	. . . . . 6 ⊢ (𝑦 = ( M ‘𝑥) → (𝑦 ⊆ No ↔ ( M ‘𝑥) ⊆ No ))
7	5, 6	syl5ibrcom 246	. . . . 5 ⊢ (𝑥 ∈ On → (𝑦 = ( M ‘𝑥) → 𝑦 ⊆ No ))
8	7	rexlimiv 3148	. . . 4 ⊢ (∃𝑥 ∈ On 𝑦 = ( M ‘𝑥) → 𝑦 ⊆ No )
9		vex 3478	. . . . 5 ⊢ 𝑦 ∈ V
10		eqeq1 2736	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑧 = ( M ‘𝑥) ↔ 𝑦 = ( M ‘𝑥)))
11	10	rexbidv 3178	. . . . 5 ⊢ (𝑧 = 𝑦 → (∃𝑥 ∈ On 𝑧 = ( M ‘𝑥) ↔ ∃𝑥 ∈ On 𝑦 = ( M ‘𝑥)))
12		fnrnfv 6948	. . . . . 6 ⊢ ( M Fn On → ran M = {𝑧 ∣ ∃𝑥 ∈ On 𝑧 = ( M ‘𝑥)})
13	2, 12	ax-mp 5	. . . . 5 ⊢ ran M = {𝑧 ∣ ∃𝑥 ∈ On 𝑧 = ( M ‘𝑥)}
14	9, 11, 13	elab2 3671	. . . 4 ⊢ (𝑦 ∈ ran M ↔ ∃𝑥 ∈ On 𝑦 = ( M ‘𝑥))
15		velpw 4606	. . . 4 ⊢ (𝑦 ∈ 𝒫 No ↔ 𝑦 ⊆ No )
16	8, 14, 15	3imtr4i 291	. . 3 ⊢ (𝑦 ∈ ran M → 𝑦 ∈ 𝒫 No )
17	16	ssriv 3985	. 2 ⊢ ran M ⊆ 𝒫 No
18		df-f 6544	. 2 ⊢ ( M :On⟶𝒫 No ↔ ( M Fn On ∧ ran M ⊆ 𝒫 No ))
19	2, 17, 18	mpbir2an 709	1 ⊢ M :On⟶𝒫 No

Syntax hints:   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2709  ∃wrex 3070  {crab 3432  Vcvv 3474   ⊆ wss 3947  𝒫 cpw 4601  ∪ cuni 4907   class class class wbr 5147   ↦ cmpt 5230   × cxp 5673  ran crn 5676   “ cima 5678  Oncon0 6361   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7405   No csur 27132   <<s csslt 27271   |s cscut 27273   M cmade 27326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-1o 8462  df-2o 8463  df-no 27135  df-slt 27136  df-bday 27137  df-sslt 27272  df-scut 27274  df-made 27331

This theorem is referenced by:  oldf  27341  newf  27342  madessno  27344  elmade  27351  elold  27353  old1  27359  madess  27360  madeoldsuc  27368  madebdayim  27371