Theorem oldf 33635

Description: The older function is a function from ordinals to sets of surreals. (Contributed by Scott Fenton, 6-Aug-2024.)

Assertion

Ref	Expression
oldf	⊢ O :On⟶𝒫 No

Proof of Theorem oldf

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-old 33626	. 2 ⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥))
2		imassrn 5917	. . . . . . . 8 ⊢ ( M “ 𝑥) ⊆ ran M
3		madef 33634	. . . . . . . . 9 ⊢ M :On⟶𝒫 No
4		frn 6509	. . . . . . . . 9 ⊢ ( M :On⟶𝒫 No → ran M ⊆ 𝒫 No )
5	3, 4	ax-mp 5	. . . . . . . 8 ⊢ ran M ⊆ 𝒫 No
6	2, 5	sstri 3903	. . . . . . 7 ⊢ ( M “ 𝑥) ⊆ 𝒫 No
7	6	sseli 3890	. . . . . 6 ⊢ (𝑦 ∈ ( M “ 𝑥) → 𝑦 ∈ 𝒫 No )
8	7	elpwid 4508	. . . . 5 ⊢ (𝑦 ∈ ( M “ 𝑥) → 𝑦 ⊆ No )
9	8	rgen 3080	. . . 4 ⊢ ∀𝑦 ∈ ( M “ 𝑥)𝑦 ⊆ No
10	9	a1i 11	. . 3 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ ( M “ 𝑥)𝑦 ⊆ No )
11		ffun 6506	. . . . . . . 8 ⊢ ( M :On⟶𝒫 No → Fun M )
12	3, 11	ax-mp 5	. . . . . . 7 ⊢ Fun M
13		vex 3413	. . . . . . . 8 ⊢ 𝑥 ∈ V
14	13	funimaex 6427	. . . . . . 7 ⊢ (Fun M → ( M “ 𝑥) ∈ V)
15	12, 14	ax-mp 5	. . . . . 6 ⊢ ( M “ 𝑥) ∈ V
16	15	uniex 7471	. . . . 5 ⊢ ∪ ( M “ 𝑥) ∈ V
17	16	elpw 4501	. . . 4 ⊢ (∪ ( M “ 𝑥) ∈ 𝒫 No ↔ ∪ ( M “ 𝑥) ⊆ No )
18		unissb 4835	. . . 4 ⊢ (∪ ( M “ 𝑥) ⊆ No ↔ ∀𝑦 ∈ ( M “ 𝑥)𝑦 ⊆ No )
19	17, 18	bitri 278	. . 3 ⊢ (∪ ( M “ 𝑥) ∈ 𝒫 No ↔ ∀𝑦 ∈ ( M “ 𝑥)𝑦 ⊆ No )
20	10, 19	sylibr 237	. 2 ⊢ (𝑥 ∈ On → ∪ ( M “ 𝑥) ∈ 𝒫 No )
21	1, 20	fmpti 6873	1 ⊢ O :On⟶𝒫 No

Syntax hints:   ∈ wcel 2111  ∀wral 3070  Vcvv 3409   ⊆ wss 3860  𝒫 cpw 4497  ∪ cuni 4801  ran crn 5529   “ cima 5531  Oncon0 6174  Fun wfun 6334  ⟶wf 6336   No csur 33440   M cmade 33620   O cold 33621

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  df-old 33626

This theorem is referenced by:  leftf  33639  rightf  33640  leftssno  33654  rightssno  33655  oldlim  33660  lrold  33668  lrrecpred  33683