Theorem bdayiun 28310

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 28299	. . . 4 ⊢ (𝐴 ∈ No → ((L‘𝐴) |s (R‘𝐴)) = 𝐴)
2	1	fveq2d 6894	. . 3 ⊢ (𝐴 ∈ No → (bday‘((L‘𝐴) |s (R‘𝐴))) = (bday‘𝐴))
3		lltr 28257	. . . 4 ⊢ (L‘𝐴) <<s (R‘𝐴)
4		fvex 6903	. . . . 5 ⊢ (O‘(bday‘𝐴)) ∈ V
5		bdayon 28147	. . . . . . 7 ⊢ (bday‘𝑥) ∈ On
6	5	onsuci 7854	. . . . . 6 ⊢ suc (bday‘𝑥) ∈ On
7	6	rgenw 3080	. . . . 5 ⊢ ∀𝑥 ∈ (O‘(bday‘𝐴))suc (bday‘𝑥) ∈ On
8		iunon 8354	. . . . 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 28292	. . . . . 6 ⊢ ((L‘𝐴) ∪ (R‘𝐴)) = (O‘(bday‘𝐴))
11	10	imaeq2i 6052	. . . . 5 ⊢ (bday “ ((L‘𝐴) ∪ (R‘𝐴))) = (bday “ (O‘(bday‘𝐴)))
12		nfv 1946	. . . . . 6 ⊢ Ⅎ𝑦 𝐴 ∈ No
13		bdayfun 28142	. . . . . . 7 ⊢ Fun bday
14	13	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ No → Fun bday)
15		fvex 6903	. . . . . . . . . 10 ⊢ (bday‘𝑦) ∈ V
16	15	sucid 6450	. . . . . . . . 9 ⊢ (bday‘𝑦) ∈ suc (bday‘𝑦)
17		fveq2 6890	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (bday‘𝑥) = (bday‘𝑦))
18	17	suceqd 6433	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → suc (bday‘𝑥) = suc (bday‘𝑦))
19	18	eleq2d 2846	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((bday‘𝑦) ∈ suc (bday‘𝑥) ↔ (bday‘𝑦) ∈ suc (bday‘𝑦)))
20	19	rspcev 3576	. . . . . . . . 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 4957	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (O‘(bday‘𝐴))) → (bday‘𝑦) ∈ ∪ 𝑥 ∈ (O‘(bday‘𝐴))suc (bday‘𝑥))
24	12, 14, 23	funimassd 6956	. . . . 5 ⊢ (𝐴 ∈ No → (bday “ (O‘(bday‘𝐴))) ⊆ ∪ 𝑥 ∈ (O‘(bday‘𝐴))suc (bday‘𝑥))
25	11, 24	eqsstrid 3947	. . . 4 ⊢ (𝐴 ∈ No → (bday “ ((L‘𝐴) ∪ (R‘𝐴))) ⊆ ∪ 𝑥 ∈ (O‘(bday‘𝐴))suc (bday‘𝑥))
26		cutbdaybnd 28190	. . . 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 3944	. 2 ⊢ (𝐴 ∈ No → (bday‘𝐴) ⊆ ∪ 𝑥 ∈ (O‘(bday‘𝐴))suc (bday‘𝑥))
29		oldbdayim 28284	. . . . 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 28147	. . . . 5 ⊢ (bday‘𝐴) ∈ On
32	5, 31	onsucssi 7856	. . . 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 5008	. 2 ⊢ (𝐴 ∈ No → ∪ 𝑥 ∈ (O‘(bday‘𝐴))suc (bday‘𝑥) ⊆ (bday‘𝐴))
35	28, 34	eqssd 3926	1 ⊢ (𝐴 ∈ No → (bday‘𝐴) = ∪ 𝑥 ∈ (O‘(bday‘𝐴))suc (bday‘𝑥))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3896   ∪ cun 4148  ∪ ciun 4950   class class class wbr 5102   “ cima 5654  Oncon0 6365  suc csuc 6367  Fun wfun 6537  ‘cfv 6543  (class class class)co 7424  Nocsur 28006  bdaycbday 28008   <<s cslts 28152   |s ccuts 28154  Ocold 28218  Lcleft 28220  Rcright 28221

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-1o 8483  df-2o 8484  df-no 28009  df-lts 28010  df-bday 28011  df-slts 28153  df-cuts 28155  df-made 28222  df-old 28223  df-left 28225  df-right 28226

This theorem is used by:  bdayle  28311