Theorem bdayiun 28005

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 27994	. . . 4 ⊢ (𝐴 ∈ No → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
2	1	fveq2d 6871	. . 3 ⊢ (𝐴 ∈ No → ( bday ‘(( L ‘𝐴) |s ( R ‘𝐴))) = ( bday ‘𝐴))
3		lltr 27952	. . . 4 ⊢ ( L ‘𝐴) <<s ( R ‘𝐴)
4		fvex 6880	. . . . 5 ⊢ ( O ‘( bday ‘𝐴)) ∈ V
5		bdayon 27842	. . . . . . 7 ⊢ ( bday ‘𝑥) ∈ On
6	5	onsuci 7819	. . . . . 6 ⊢ suc ( bday ‘𝑥) ∈ On
7	6	rgenw 3080	. . . . 5 ⊢ ∀𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥) ∈ On
8		iunon 8310	. . . . 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 702	. . . 4 ⊢ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥) ∈ On
10		lrold 27987	. . . . . 6 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴))
11	10	imaeq2i 6047	. . . . 5 ⊢ ( bday “ (( L ‘𝐴) ∪ ( R ‘𝐴))) = ( bday “ ( O ‘( bday ‘𝐴)))
12		nfv 1934	. . . . . 6 ⊢ Ⅎ𝑦 𝐴 ∈ No
13		bdayfun 27837	. . . . . . 7 ⊢ Fun bday
14	13	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ No → Fun bday )
15		fvex 6880	. . . . . . . . . 10 ⊢ ( bday ‘𝑦) ∈ V
16	15	sucid 6430	. . . . . . . . 9 ⊢ ( bday ‘𝑦) ∈ suc ( bday ‘𝑦)
17		fveq2 6867	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ( bday ‘𝑥) = ( bday ‘𝑦))
18	17	suceqd 6413	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → suc ( bday ‘𝑥) = suc ( bday ‘𝑦))
19	18	eleq2d 2848	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (( bday ‘𝑦) ∈ suc ( bday ‘𝑥) ↔ ( bday ‘𝑦) ∈ suc ( bday ‘𝑦)))
20	19	rspcev 3581	. . . . . . . . 9 ⊢ ((𝑦 ∈ ( O ‘( bday ‘𝐴)) ∧ ( bday ‘𝑦) ∈ suc ( bday ‘𝑦)) → ∃𝑥 ∈ ( O ‘( bday ‘𝐴))( bday ‘𝑦) ∈ suc ( bday ‘𝑥))
21	16, 20	mpan2 701	. . . . . . . 8 ⊢ (𝑦 ∈ ( O ‘( bday ‘𝐴)) → ∃𝑥 ∈ ( O ‘( bday ‘𝐴))( bday ‘𝑦) ∈ suc ( bday ‘𝑥))
22	21	adantl 485	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( O ‘( bday ‘𝐴))) → ∃𝑥 ∈ ( O ‘( bday ‘𝐴))( bday ‘𝑦) ∈ suc ( bday ‘𝑥))
23	22	eliund 4956	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( O ‘( bday ‘𝐴))) → ( bday ‘𝑦) ∈ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
24	12, 14, 23	funimassd 6933	. . . . 5 ⊢ (𝐴 ∈ No → ( bday “ ( O ‘( bday ‘𝐴))) ⊆ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
25	11, 24	eqsstrid 3974	. . . 4 ⊢ (𝐴 ∈ No → ( bday “ (( L ‘𝐴) ∪ ( R ‘𝐴))) ⊆ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
26		cutbdaybnd 27885	. . . 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 1486	. . 3 ⊢ (𝐴 ∈ No → ( bday ‘(( L ‘𝐴) |s ( R ‘𝐴))) ⊆ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
28	2, 27	eqsstrrd 3971	. 2 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) ⊆ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
29		oldbdayim 27979	. . . . 5 ⊢ (𝑥 ∈ ( O ‘( bday ‘𝐴)) → ( bday ‘𝑥) ∈ ( bday ‘𝐴))
30	29	adantl 485	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( O ‘( bday ‘𝐴))) → ( bday ‘𝑥) ∈ ( bday ‘𝐴))
31		bdayon 27842	. . . . 5 ⊢ ( bday ‘𝐴) ∈ On
32	5, 31	onsucssi 7821	. . . 4 ⊢ (( bday ‘𝑥) ∈ ( bday ‘𝐴) ↔ suc ( bday ‘𝑥) ⊆ ( bday ‘𝐴))
33	30, 32	sylib 220	. . 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 3953	1 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086  Vcvv 3454   ∪ cun 3902   ⊆ wss 3904  ∪ ciun 4949   class class class wbr 5100   “ cima 5650  Oncon0 6346  suc csuc 6348  Fun wfun 6515  ‘cfv 6521  (class class class)co 7396   No csur 27701   bday cbday 27703   <<s cslts 27847   |s ccuts 27849   O cold 27913   L cleft 27915   R cright 27916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-1o 8437  df-2o 8438  df-no 27704  df-lts 27705  df-bday 27706  df-slts 27848  df-cuts 27850  df-made 27917  df-old 27918  df-left 27920  df-right 27921

This theorem is referenced by:  bdayle  28006