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.

Step	Hyp	Ref	Expression
1		df-made 27887	. . 3 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))))
2	1	tfr1 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
5	3, 4	eqsstrdi 4027	. . . . . 6 ⊢ (𝑥 ∈ On → ( M ‘𝑥) ⊆ No )
6		sseq1 4008	. . . . . 6 ⊢ (𝑦 = ( M ‘𝑥) → (𝑦 ⊆ No ↔ ( M ‘𝑥) ⊆ No ))
7	5, 6	syl5ibrcom 247	. . . . 5 ⊢ (𝑥 ∈ On → (𝑦 = ( M ‘𝑥) → 𝑦 ⊆ No ))
8	7	rexlimiv 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)))
11	10	rexbidv 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)})
13	2, 12	ax-mp 5	. . . . 5 ⊢ ran M = {𝑧 ∣ ∃𝑥 ∈ On 𝑧 = ( M ‘𝑥)}
14	9, 11, 13	elab2 3681	. . . 4 ⊢ (𝑦 ∈ ran M ↔ ∃𝑥 ∈ On 𝑦 = ( M ‘𝑥))
15		velpw 4604	. . . 4 ⊢ (𝑦 ∈ 𝒫 No ↔ 𝑦 ⊆ No )
16	8, 14, 15	3imtr4i 292	. . 3 ⊢ (𝑦 ∈ ran M → 𝑦 ∈ 𝒫 No )
17	16	ssriv 3986	. 2 ⊢ ran M ⊆ 𝒫 No
18		df-f 6564	. 2 ⊢ ( M :On⟶𝒫 No ↔ ( M Fn On ∧ ran M ⊆ 𝒫 No ))
19	2, 17, 18	mpbir2an 711	1 ⊢ M :On⟶𝒫 No

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