Theorem madef 28227

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 28218	. . 3 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))))
2	1	tfr1 8408	. 2 ⊢ M Fn On
3		madeval2 28224	. . . . . . 7 ⊢ (𝑥 ∈ On → (M‘𝑥) = {𝑦 ∈ No ∣ ∃𝑧 ∈ 𝒫 ∪ (M “ 𝑥)∃𝑤 ∈ 𝒫 ∪ (M “ 𝑥)(𝑧 <<s 𝑤 ∧ (𝑧 |s 𝑤) = 𝑦)})
4		ssrab2 4005	. . . . . . 7 ⊢ {𝑦 ∈ No ∣ ∃𝑧 ∈ 𝒫 ∪ (M “ 𝑥)∃𝑤 ∈ 𝒫 ∪ (M “ 𝑥)(𝑧 <<s 𝑤 ∧ (𝑧 |s 𝑤) = 𝑦)} ⊆ No
5	3, 4	eqsstrdi 3952	. . . . . 6 ⊢ (𝑥 ∈ On → (M‘𝑥) ⊆ No)
6		sseq1 3933	. . . . . 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 3156	. . . 4 ⊢ (∃𝑥 ∈ On 𝑦 = (M‘𝑥) → 𝑦 ⊆ No)
9		vex 3454	. . . . 5 ⊢ 𝑦 ∈ V
10		eqeq1 2764	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑧 = (M‘𝑥) ↔ 𝑦 = (M‘𝑥)))
11	10	rexbidv 3186	. . . . 5 ⊢ (𝑧 = 𝑦 → (∃𝑥 ∈ On 𝑧 = (M‘𝑥) ↔ ∃𝑥 ∈ On 𝑦 = (M‘𝑥)))
12		fnrnfv 6944	. . . . . 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 3635	. . . 4 ⊢ (𝑦 ∈ ran M ↔ ∃𝑥 ∈ On 𝑦 = (M‘𝑥))
15		velpw 4561	. . . 4 ⊢ (𝑦 ∈ 𝒫 No ↔ 𝑦 ⊆ No)
16	8, 14, 15	3imtr4i 295	. . 3 ⊢ (𝑦 ∈ ran M → 𝑦 ∈ 𝒫 No)
17	16	ssriv 3912	. 2 ⊢ ran M ⊆ 𝒫 No
18		df-f 6542	. 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 1569   ∈ wcel 2144  {cab 2738  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3895  𝒫 cpw 4556  ∪ cuni 4866   class class class wbr 5102   ↦ cmpt 5185   × cxp 5648  ran crn 5651   “ cima 5653  Oncon0 6362   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  Nocsur 28002   <<s cslts 28148   |s ccuts 28150  Mcmade 28213

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-no 28005  df-lts 28006  df-bday 28007  df-slts 28149  df-cuts 28151  df-made 28218

This theorem is used by:  oldf  28228  newf  28229  madessno  28231  elmade  28248  elold  28250  old1  28256  madess  28257  madeoldsuc  28276  madebdayim  28279  madefi  28304  oldfi  28305  oldfib  28768