Theorem bdayiun 27864

Description: The birthday of a surreal is the least upper bound of the successors of the birthdays of its options. This is the definition of the birthday of a combinatorial game in the Lean Combinatorial Game Theory library at https://github.com/vihdzp/combinatorial-games. (Contributed by Scott Fenton, 22-Nov-2025.)

Assertion

Ref	Expression
bdayiun	⊢ (𝐴 ∈ No → ( bday ‘𝐴) = ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))

Distinct variable group:   𝑥,𝐴

Proof of Theorem bdayiun

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lrcut 27853	. . . 4 ⊢ (𝐴 ∈ No → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
2	1	fveq2d 6844	. . 3 ⊢ (𝐴 ∈ No → ( bday ‘(( L ‘𝐴) |s ( R ‘𝐴))) = ( bday ‘𝐴))
3		lltropt 27821	. . . 4 ⊢ ( L ‘𝐴) <<s ( R ‘𝐴)
4		fvex 6853	. . . . 5 ⊢ ( O ‘( bday ‘𝐴)) ∈ V
5		bdayelon 27721	. . . . . . 7 ⊢ ( bday ‘𝑥) ∈ On
6	5	onsuci 7794	. . . . . 6 ⊢ suc ( bday ‘𝑥) ∈ On
7	6	rgenw 3048	. . . . 5 ⊢ ∀𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥) ∈ On
8		iunon 8285	. . . . 5 ⊢ ((( O ‘( bday ‘𝐴)) ∈ V ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥) ∈ On) → ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥) ∈ On)
9	4, 7, 8	mp2an 692	. . . 4 ⊢ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥) ∈ On
10		lrold 27846	. . . . . 6 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴))
11	10	imaeq2i 6018	. . . . 5 ⊢ ( bday “ (( L ‘𝐴) ∪ ( R ‘𝐴))) = ( bday “ ( O ‘( bday ‘𝐴)))
12		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑦 𝐴 ∈ No
13		bdayfun 27717	. . . . . . 7 ⊢ Fun bday
14	13	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ No → Fun bday )
15		fvex 6853	. . . . . . . . . 10 ⊢ ( bday ‘𝑦) ∈ V
16	15	sucid 6404	. . . . . . . . 9 ⊢ ( bday ‘𝑦) ∈ suc ( bday ‘𝑦)
17		fveq2 6840	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ( bday ‘𝑥) = ( bday ‘𝑦))
18	17	suceqd 6387	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → suc ( bday ‘𝑥) = suc ( bday ‘𝑦))
19	18	eleq2d 2814	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (( bday ‘𝑦) ∈ suc ( bday ‘𝑥) ↔ ( bday ‘𝑦) ∈ suc ( bday ‘𝑦)))
20	19	rspcev 3585	. . . . . . . . 9 ⊢ ((𝑦 ∈ ( O ‘( bday ‘𝐴)) ∧ ( bday ‘𝑦) ∈ suc ( bday ‘𝑦)) → ∃𝑥 ∈ ( O ‘( bday ‘𝐴))( bday ‘𝑦) ∈ suc ( bday ‘𝑥))
21	16, 20	mpan2 691	. . . . . . . 8 ⊢ (𝑦 ∈ ( O ‘( bday ‘𝐴)) → ∃𝑥 ∈ ( O ‘( bday ‘𝐴))( bday ‘𝑦) ∈ suc ( bday ‘𝑥))
22	21	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( O ‘( bday ‘𝐴))) → ∃𝑥 ∈ ( O ‘( bday ‘𝐴))( bday ‘𝑦) ∈ suc ( bday ‘𝑥))
23	22	eliund 4958	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( O ‘( bday ‘𝐴))) → ( bday ‘𝑦) ∈ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
24	12, 14, 23	funimassd 6909	. . . . 5 ⊢ (𝐴 ∈ No → ( bday “ ( O ‘( bday ‘𝐴))) ⊆ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
25	11, 24	eqsstrid 3982	. . . 4 ⊢ (𝐴 ∈ No → ( bday “ (( L ‘𝐴) ∪ ( R ‘𝐴))) ⊆ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
26		scutbdaybnd 27761	. . . 4 ⊢ ((( L ‘𝐴) <<s ( R ‘𝐴) ∧ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥) ∈ On ∧ ( bday “ (( L ‘𝐴) ∪ ( R ‘𝐴))) ⊆ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥)) → ( bday ‘(( L ‘𝐴) |s ( R ‘𝐴))) ⊆ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
27	3, 9, 25, 26	mp3an12i 1467	. . 3 ⊢ (𝐴 ∈ No → ( bday ‘(( L ‘𝐴) |s ( R ‘𝐴))) ⊆ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
28	2, 27	eqsstrrd 3979	. 2 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) ⊆ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
29		oldbdayim 27838	. . . . 5 ⊢ (𝑥 ∈ ( O ‘( bday ‘𝐴)) → ( bday ‘𝑥) ∈ ( bday ‘𝐴))
30	29	adantl 481	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( O ‘( bday ‘𝐴))) → ( bday ‘𝑥) ∈ ( bday ‘𝐴))
31		bdayelon 27721	. . . . 5 ⊢ ( bday ‘𝐴) ∈ On
32	5, 31	onsucssi 7797	. . . 4 ⊢ (( bday ‘𝑥) ∈ ( bday ‘𝐴) ↔ suc ( bday ‘𝑥) ⊆ ( bday ‘𝐴))
33	30, 32	sylib 218	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( O ‘( bday ‘𝐴))) → suc ( bday ‘𝑥) ⊆ ( bday ‘𝐴))
34	33	iunssd 5009	. 2 ⊢ (𝐴 ∈ No → ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥) ⊆ ( bday ‘𝐴))
35	28, 34	eqssd 3961	1 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3444   ∪ cun 3909   ⊆ wss 3911  ∪ ciun 4951   class class class wbr 5102   “ cima 5634  Oncon0 6320  suc csuc 6322  Fun wfun 6493  ‘cfv 6499  (class class class)co 7369   No csur 27584   bday cbday 27586   <<s csslt 27726   |s cscut 27728   O cold 27788   L cleft 27790   R cright 27791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-1o 8411  df-2o 8412  df-no 27587  df-slt 27588  df-bday 27589  df-sslt 27727  df-scut 27729  df-made 27792  df-old 27793  df-left 27795  df-right 27796

This theorem is referenced by:  bdayle  27865