Theorem madef 33634

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 33625	. . 3 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))))
2	1	tfr1 8049	. 2 ⊢ M Fn On
3		madeval2 33631	. . . . . . 7 ⊢ (𝑥 ∈ On → ( M ‘𝑥) = {𝑦 ∈ No ∣ ∃𝑧 ∈ 𝒫 ∪ ( M “ 𝑥)∃𝑤 ∈ 𝒫 ∪ ( M “ 𝑥)(𝑧 <<s 𝑤 ∧ (𝑧 |s 𝑤) = 𝑦)})
4		ssrab2 3986	. . . . . . 7 ⊢ {𝑦 ∈ No ∣ ∃𝑧 ∈ 𝒫 ∪ ( M “ 𝑥)∃𝑤 ∈ 𝒫 ∪ ( M “ 𝑥)(𝑧 <<s 𝑤 ∧ (𝑧 |s 𝑤) = 𝑦)} ⊆ No
5	3, 4	eqsstrdi 3948	. . . . . 6 ⊢ (𝑥 ∈ On → ( M ‘𝑥) ⊆ No )
6		sseq1 3919	. . . . . 6 ⊢ (𝑦 = ( M ‘𝑥) → (𝑦 ⊆ No ↔ ( M ‘𝑥) ⊆ No ))
7	5, 6	syl5ibrcom 250	. . . . 5 ⊢ (𝑥 ∈ On → (𝑦 = ( M ‘𝑥) → 𝑦 ⊆ No ))
8	7	rexlimiv 3204	. . . 4 ⊢ (∃𝑥 ∈ On 𝑦 = ( M ‘𝑥) → 𝑦 ⊆ No )
9		vex 3413	. . . . 5 ⊢ 𝑦 ∈ V
10		eqeq1 2762	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑧 = ( M ‘𝑥) ↔ 𝑦 = ( M ‘𝑥)))
11	10	rexbidv 3221	. . . . 5 ⊢ (𝑧 = 𝑦 → (∃𝑥 ∈ On 𝑧 = ( M ‘𝑥) ↔ ∃𝑥 ∈ On 𝑦 = ( M ‘𝑥)))
12		fnrnfv 6718	. . . . . 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 3593	. . . 4 ⊢ (𝑦 ∈ ran M ↔ ∃𝑥 ∈ On 𝑦 = ( M ‘𝑥))
15		velpw 4502	. . . 4 ⊢ (𝑦 ∈ 𝒫 No ↔ 𝑦 ⊆ No )
16	8, 14, 15	3imtr4i 295	. . 3 ⊢ (𝑦 ∈ ran M → 𝑦 ∈ 𝒫 No )
17	16	ssriv 3898	. 2 ⊢ ran M ⊆ 𝒫 No
18		df-f 6344	. 2 ⊢ ( M :On⟶𝒫 No ↔ ( M Fn On ∧ ran M ⊆ 𝒫 No ))
19	2, 17, 18	mpbir2an 710	1 ⊢ M :On⟶𝒫 No

Syntax hints:   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  ∃wrex 3071  {crab 3074  Vcvv 3409   ⊆ wss 3860  𝒫 cpw 4497  ∪ cuni 4801   class class class wbr 5036   ↦ cmpt 5116   × cxp 5526  ran crn 5529   “ cima 5531  Oncon0 6174   Fn wfn 6335  ⟶wf 6336  ‘cfv 6340  (class class class)co 7156   No csur 33440   <<s csslt 33572   |s cscut 33574   M cmade 33620

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-wrecs 7963  df-recs 8024  df-1o 8118  df-2o 8119  df-no 33443  df-slt 33444  df-bday 33445  df-sslt 33573  df-scut 33575  df-made 33625

This theorem is referenced by:  oldf  33635  newf  33636  elmade  33641  elold  33643  madeoldsuc  33658