Theorem bdayiun 27887

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 27876	. . . 4 ⊢ (𝐴 ∈ No → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
2	1	fveq2d 6836	. . 3 ⊢ (𝐴 ∈ No → ( bday ‘(( L ‘𝐴) |s ( R ‘𝐴))) = ( bday ‘𝐴))
3		lltropt 27844	. . . 4 ⊢ ( L ‘𝐴) <<s ( R ‘𝐴)
4		fvex 6845	. . . . 5 ⊢ ( O ‘( bday ‘𝐴)) ∈ V
5		bdayelon 27742	. . . . . . 7 ⊢ ( bday ‘𝑥) ∈ On
6	5	onsuci 7779	. . . . . 6 ⊢ suc ( bday ‘𝑥) ∈ On
7	6	rgenw 3053	. . . . 5 ⊢ ∀𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥) ∈ On
8		iunon 8269	. . . . 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 27869	. . . . . 6 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴))
11	10	imaeq2i 6015	. . . . 5 ⊢ ( bday “ (( L ‘𝐴) ∪ ( R ‘𝐴))) = ( bday “ ( O ‘( bday ‘𝐴)))
12		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑦 𝐴 ∈ No
13		bdayfun 27738	. . . . . . 7 ⊢ Fun bday
14	13	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ No → Fun bday )
15		fvex 6845	. . . . . . . . . 10 ⊢ ( bday ‘𝑦) ∈ V
16	15	sucid 6399	. . . . . . . . 9 ⊢ ( bday ‘𝑦) ∈ suc ( bday ‘𝑦)
17		fveq2 6832	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ( bday ‘𝑥) = ( bday ‘𝑦))
18	17	suceqd 6382	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → suc ( bday ‘𝑥) = suc ( bday ‘𝑦))
19	18	eleq2d 2820	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (( bday ‘𝑦) ∈ suc ( bday ‘𝑥) ↔ ( bday ‘𝑦) ∈ suc ( bday ‘𝑦)))
20	19	rspcev 3574	. . . . . . . . 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 4951	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( O ‘( bday ‘𝐴))) → ( bday ‘𝑦) ∈ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
24	12, 14, 23	funimassd 6898	. . . . 5 ⊢ (𝐴 ∈ No → ( bday “ ( O ‘( bday ‘𝐴))) ⊆ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
25	11, 24	eqsstrid 3970	. . . 4 ⊢ (𝐴 ∈ No → ( bday “ (( L ‘𝐴) ∪ ( R ‘𝐴))) ⊆ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
26		scutbdaybnd 27783	. . . 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 3967	. 2 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) ⊆ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
29		oldbdayim 27861	. . . . 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 27742	. . . . 5 ⊢ ( bday ‘𝐴) ∈ On
32	5, 31	onsucssi 7781	. . . 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 5004	. 2 ⊢ (𝐴 ∈ No → ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥) ⊆ ( bday ‘𝐴))
35	28, 34	eqssd 3949	1 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058  Vcvv 3438   ∪ cun 3897   ⊆ wss 3899  ∪ ciun 4944   class class class wbr 5096   “ cima 5625  Oncon0 6315  suc csuc 6317  Fun wfun 6484  ‘cfv 6490  (class class class)co 7356   No csur 27605   bday cbday 27607   <<s csslt 27747   |s cscut 27749   O cold 27811   L cleft 27813   R cright 27814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-1o 8395  df-2o 8396  df-no 27608  df-slt 27609  df-bday 27610  df-sslt 27748  df-scut 27750  df-made 27815  df-old 27816  df-left 27818  df-right 27819

This theorem is referenced by:  bdayle  27888