Theorem bdayiun 28159

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 28148	. . . 4 ⊢ (𝐴 ∈ No → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
2	1	fveq2d 6889	. . 3 ⊢ (𝐴 ∈ No → ( bday ‘(( L ‘𝐴) |s ( R ‘𝐴))) = ( bday ‘𝐴))
3		lltr 28106	. . . 4 ⊢ ( L ‘𝐴) <<s ( R ‘𝐴)
4		fvex 6898	. . . . 5 ⊢ ( O ‘( bday ‘𝐴)) ∈ V
5		bdayon 27996	. . . . . . 7 ⊢ ( bday ‘𝑥) ∈ On
6	5	onsuci 7841	. . . . . 6 ⊢ suc ( bday ‘𝑥) ∈ On
7	6	rgenw 3085	. . . . 5 ⊢ ∀𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥) ∈ On
8		iunon 8332	. . . . 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 705	. . . 4 ⊢ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥) ∈ On
10		lrold 28141	. . . . . 6 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴))
11	10	imaeq2i 6062	. . . . 5 ⊢ ( bday “ (( L ‘𝐴) ∪ ( R ‘𝐴))) = ( bday “ ( O ‘( bday ‘𝐴)))
12		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑦 𝐴 ∈ No
13		bdayfun 27991	. . . . . . 7 ⊢ Fun bday
14	13	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ No → Fun bday )
15		fvex 6898	. . . . . . . . . 10 ⊢ ( bday ‘𝑦) ∈ V
16	15	sucid 6449	. . . . . . . . 9 ⊢ ( bday ‘𝑦) ∈ suc ( bday ‘𝑦)
17		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ( bday ‘𝑥) = ( bday ‘𝑦))
18	17	suceqd 6432	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → suc ( bday ‘𝑥) = suc ( bday ‘𝑦))
19	18	eleq2d 2851	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (( bday ‘𝑦) ∈ suc ( bday ‘𝑥) ↔ ( bday ‘𝑦) ∈ suc ( bday ‘𝑦)))
20	19	rspcev 3583	. . . . . . . . 9 ⊢ ((𝑦 ∈ ( O ‘( bday ‘𝐴)) ∧ ( bday ‘𝑦) ∈ suc ( bday ‘𝑦)) → ∃𝑥 ∈ ( O ‘( bday ‘𝐴))( bday ‘𝑦) ∈ suc ( bday ‘𝑥))
21	16, 20	mpan2 704	. . . . . . . 8 ⊢ (𝑦 ∈ ( O ‘( bday ‘𝐴)) → ∃𝑥 ∈ ( O ‘( bday ‘𝐴))( bday ‘𝑦) ∈ suc ( bday ‘𝑥))
22	21	adantl 487	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( O ‘( bday ‘𝐴))) → ∃𝑥 ∈ ( O ‘( bday ‘𝐴))( bday ‘𝑦) ∈ suc ( bday ‘𝑥))
23	22	eliund 4965	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( O ‘( bday ‘𝐴))) → ( bday ‘𝑦) ∈ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
24	12, 14, 23	funimassd 6951	. . . . 5 ⊢ (𝐴 ∈ No → ( bday “ ( O ‘( bday ‘𝐴))) ⊆ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
25	11, 24	eqsstrid 3976	. . . 4 ⊢ (𝐴 ∈ No → ( bday “ (( L ‘𝐴) ∪ ( R ‘𝐴))) ⊆ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
26		cutbdaybnd 28039	. . . 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 1494	. . 3 ⊢ (𝐴 ∈ No → ( bday ‘(( L ‘𝐴) |s ( R ‘𝐴))) ⊆ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
28	2, 27	eqsstrrd 3973	. 2 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) ⊆ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
29		oldbdayim 28133	. . . . 5 ⊢ (𝑥 ∈ ( O ‘( bday ‘𝐴)) → ( bday ‘𝑥) ∈ ( bday ‘𝐴))
30	29	adantl 487	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( O ‘( bday ‘𝐴))) → ( bday ‘𝑥) ∈ ( bday ‘𝐴))
31		bdayon 27996	. . . . 5 ⊢ ( bday ‘𝐴) ∈ On
32	5, 31	onsucssi 7843	. . . 4 ⊢ (( bday ‘𝑥) ∈ ( bday ‘𝐴) ↔ suc ( bday ‘𝑥) ⊆ ( bday ‘𝐴))
33	30, 32	sylib 221	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( O ‘( bday ‘𝐴))) → suc ( bday ‘𝑥) ⊆ ( bday ‘𝐴))
34	33	iunssd 5017	. 2 ⊢ (𝐴 ∈ No → ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥) ⊆ ( bday ‘𝐴))
35	28, 34	eqssd 3955	1 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ∪ cun 3904   ⊆ wss 3906  ∪ ciun 4958   class class class wbr 5111   “ cima 5666  Oncon0 6364  suc csuc 6366  Fun wfun 6534  ‘cfv 6540  (class class class)co 7419   No csur 27855   bday cbday 27857   <<s cslts 28001   |s ccuts 28003   O cold 28067   L cleft 28069   R cright 28070

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-1o 8459  df-2o 8460  df-no 27858  df-lts 27859  df-bday 27860  df-slts 28002  df-cuts 28004  df-made 28071  df-old 28072  df-left 28074  df-right 28075

This theorem is used by:  bdayle  28160