Theorem bdayiun 28358

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 28347	. . . 4 ⊢ (𝐴 ∈ No → ((L‘𝐴) |s (R‘𝐴)) = 𝐴)
2	1	fveq2d 6920	. . 3 ⊢ (𝐴 ∈ No → (bday‘((L‘𝐴) |s (R‘𝐴))) = (bday‘𝐴))
3		lltr 28305	. . . 4 ⊢ (L‘𝐴) <<s (R‘𝐴)
4		fvex 6929	. . . . 5 ⊢ (O‘(bday‘𝐴)) ∈ V
5		bdayon 28195	. . . . . . 7 ⊢ (bday‘𝑥) ∈ On
6	5	onsuci 7884	. . . . . 6 ⊢ suc (bday‘𝑥) ∈ On
7	6	rgenw 3084	. . . . 5 ⊢ ∀𝑥 ∈ (O‘(bday‘𝐴))suc (bday‘𝑥) ∈ On
8		iunon 8385	. . . . 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 28340	. . . . . 6 ⊢ ((L‘𝐴) ∪ (R‘𝐴)) = (O‘(bday‘𝐴))
11	10	imaeq2i 6068	. . . . 5 ⊢ (bday “ ((L‘𝐴) ∪ (R‘𝐴))) = (bday “ (O‘(bday‘𝐴)))
12		nfv 1949	. . . . . 6 ⊢ Ⅎ𝑦 𝐴 ∈ No
13		bdayfun 28190	. . . . . . 7 ⊢ Fun bday
14	13	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ No → Fun bday)
15		fvex 6929	. . . . . . . . . 10 ⊢ (bday‘𝑦) ∈ V
16	15	sucid 6473	. . . . . . . . 9 ⊢ (bday‘𝑦) ∈ suc (bday‘𝑦)
17		fveq2 6916	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (bday‘𝑥) = (bday‘𝑦))
18	17	suceqd 6456	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → suc (bday‘𝑥) = suc (bday‘𝑦))
19	18	eleq2d 2850	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((bday‘𝑦) ∈ suc (bday‘𝑥) ↔ (bday‘𝑦) ∈ suc (bday‘𝑦)))
20	19	rspcev 3580	. . . . . . . . 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 4963	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (O‘(bday‘𝐴))) → (bday‘𝑦) ∈ ∪ 𝑥 ∈ (O‘(bday‘𝐴))suc (bday‘𝑥))
24	12, 14, 23	funimassd 6982	. . . . 5 ⊢ (𝐴 ∈ No → (bday “ (O‘(bday‘𝐴))) ⊆ ∪ 𝑥 ∈ (O‘(bday‘𝐴))suc (bday‘𝑥))
25	11, 24	eqsstrid 3951	. . . 4 ⊢ (𝐴 ∈ No → (bday “ ((L‘𝐴) ∪ (R‘𝐴))) ⊆ ∪ 𝑥 ∈ (O‘(bday‘𝐴))suc (bday‘𝑥))
26		cutbdaybnd 28238	. . . 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 1495	. . 3 ⊢ (𝐴 ∈ No → (bday‘((L‘𝐴) |s (R‘𝐴))) ⊆ ∪ 𝑥 ∈ (O‘(bday‘𝐴))suc (bday‘𝑥))
28	2, 27	eqsstrrd 3948	. 2 ⊢ (𝐴 ∈ No → (bday‘𝐴) ⊆ ∪ 𝑥 ∈ (O‘(bday‘𝐴))suc (bday‘𝑥))
29		oldbdayim 28332	. . . . 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 28195	. . . . 5 ⊢ (bday‘𝐴) ∈ On
32	5, 31	onsucssi 7886	. . . 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 5014	. 2 ⊢ (𝐴 ∈ No → ∪ 𝑥 ∈ (O‘(bday‘𝐴))suc (bday‘𝑥) ⊆ (bday‘𝐴))
35	28, 34	eqssd 3930	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 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∪ cun 4152  ∪ ciun 4956   class class class wbr 5108   “ cima 5661  Oncon0 6388  suc csuc 6390  Fun wfun 6560  ‘cfv 6566  (class class class)co 7451  Nocsur 28054  bdaycbday 28056   <<s cslts 28200   |s ccuts 28202  Ocold 28266  Lcleft 28268  Rcright 28269

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-1o 8514  df-2o 8515  df-no 28057  df-lts 28058  df-bday 28059  df-slts 28201  df-cuts 28203  df-made 28270  df-old 28271  df-left 28273  df-right 28274

This theorem is used by:  bdayle  28359