Theorem madef 28204

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 28195	. . 3 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))))
2	1	tfr1 8389	. 2 ⊢ M Fn On
3		madeval2 28201	. . . . . . 7 ⊢ (𝑥 ∈ On → ( M ‘𝑥) = {𝑦 ∈ No ∣ ∃𝑧 ∈ 𝒫 ∪ ( M “ 𝑥)∃𝑤 ∈ 𝒫 ∪ ( M “ 𝑥)(𝑧 <<s 𝑤 ∧ (𝑧 |s 𝑤) = 𝑦)})
4		ssrab2 4028	. . . . . . 7 ⊢ {𝑦 ∈ No ∣ ∃𝑧 ∈ 𝒫 ∪ ( M “ 𝑥)∃𝑤 ∈ 𝒫 ∪ ( M “ 𝑥)(𝑧 <<s 𝑤 ∧ (𝑧 |s 𝑤) = 𝑦)} ⊆ No
5	3, 4	eqsstrdi 3975	. . . . . 6 ⊢ (𝑥 ∈ On → ( M ‘𝑥) ⊆ No )
6		sseq1 3956	. . . . . 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 3157	. . . 4 ⊢ (∃𝑥 ∈ On 𝑦 = ( M ‘𝑥) → 𝑦 ⊆ No )
9		vex 3455	. . . . 5 ⊢ 𝑦 ∈ V
10		eqeq1 2765	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑧 = ( M ‘𝑥) ↔ 𝑦 = ( M ‘𝑥)))
11	10	rexbidv 3187	. . . . 5 ⊢ (𝑧 = 𝑦 → (∃𝑥 ∈ On 𝑧 = ( M ‘𝑥) ↔ ∃𝑥 ∈ On 𝑦 = ( M ‘𝑥)))
12		fnrnfv 6936	. . . . . 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 3636	. . . 4 ⊢ (𝑦 ∈ ran M ↔ ∃𝑥 ∈ On 𝑦 = ( M ‘𝑥))
15		velpw 4562	. . . 4 ⊢ (𝑦 ∈ 𝒫 No ↔ 𝑦 ⊆ No )
16	8, 14, 15	3imtr4i 295	. . 3 ⊢ (𝑦 ∈ ran M → 𝑦 ∈ 𝒫 No )
17	16	ssriv 3935	. 2 ⊢ ran M ⊆ 𝒫 No
18		df-f 6535	. 2 ⊢ ( M :On⟶𝒫 No ↔ ( M Fn On ∧ ran M ⊆ 𝒫 No ))
19	2, 17, 18	mpbir2an 724	1 ⊢ M :On⟶𝒫 No

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2739  ∃wrex 3087  {crab 3413  Vcvv 3451   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867   class class class wbr 5103   ↦ cmpt 5186   × cxp 5649  ran crn 5652   “ cima 5654  Oncon0 6355   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412   No csur 27979   <<s cslts 28125   |s ccuts 28127   M cmade 28190

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-1o 8460  df-2o 8461  df-no 27982  df-lts 27983  df-bday 27984  df-slts 28126  df-cuts 28128  df-made 28195

This theorem is used by:  oldf  28205  newf  28206  madessno  28208  elmade  28225  elold  28227  old1  28233  madess  28234  madeoldsuc  28253  madebdayim  28256  madefi  28281  oldfi  28282  oldfib  28745