Theorem madef 28252

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 28243	. . 3 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))))
2	1	tfr1 8433	. 2 ⊢ M Fn On
3		madeval2 28249	. . . . . . 7 ⊢ (𝑥 ∈ On → (M‘𝑥) = {𝑦 ∈ No ∣ ∃𝑧 ∈ 𝒫 ∪ (M “ 𝑥)∃𝑤 ∈ 𝒫 ∪ (M “ 𝑥)(𝑧 <<s 𝑤 ∧ (𝑧 |s 𝑤) = 𝑦)})
4		ssrab2 4008	. . . . . . 7 ⊢ {𝑦 ∈ No ∣ ∃𝑧 ∈ 𝒫 ∪ (M “ 𝑥)∃𝑤 ∈ 𝒫 ∪ (M “ 𝑥)(𝑧 <<s 𝑤 ∧ (𝑧 |s 𝑤) = 𝑦)} ⊆ No
5	3, 4	eqsstrdi 3957	. . . . . 6 ⊢ (𝑥 ∈ On → (M‘𝑥) ⊆ No)
6		sseq1 3938	. . . . . 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 3160	. . . 4 ⊢ (∃𝑥 ∈ On 𝑦 = (M‘𝑥) → 𝑦 ⊆ No)
9		vex 3458	. . . . 5 ⊢ 𝑦 ∈ V
10		eqeq1 2767	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑧 = (M‘𝑥) ↔ 𝑦 = (M‘𝑥)))
11	10	rexbidv 3190	. . . . 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 3639	. . . 4 ⊢ (𝑦 ∈ ran M ↔ ∃𝑥 ∈ On 𝑦 = (M‘𝑥))
15		velpw 4565	. . . 4 ⊢ (𝑦 ∈ 𝒫 No ↔ 𝑦 ⊆ No)
16	8, 14, 15	3imtr4i 295	. . 3 ⊢ (𝑦 ∈ ran M → 𝑦 ∈ 𝒫 No)
17	16	ssriv 3917	. 2 ⊢ ran M ⊆ 𝒫 No
18		df-f 6562	. 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 2147  {cab 2741  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900  𝒫 cpw 4560  ∪ cuni 4870   class class class wbr 5106   ↦ cmpt 5189   × cxp 5653  ran crn 5656   “ cima 5658  Oncon0 6380   Fn wfn 6553  ⟶wf 6554  ‘cfv 6558  (class class class)co 7443  Nocsur 28027   <<s cslts 28173   |s ccuts 28175  Mcmade 28238

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-1o 8504  df-2o 8505  df-no 28030  df-lts 28031  df-bday 28032  df-slts 28174  df-cuts 28176  df-made 28243

This theorem is used by:  oldf  28253  newf  28254  madessno  28256  elmade  28273  elold  28275  old1  28281  madess  28282  madeoldsuc  28301  madebdayim  28304  madefi  28329  oldfi  28330  oldfib  28793