Theorem bdayiun 27923

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 27912	. . . 4 ⊢ (𝐴 ∈ No → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
2	1	fveq2d 6846	. . 3 ⊢ (𝐴 ∈ No → ( bday ‘(( L ‘𝐴) |s ( R ‘𝐴))) = ( bday ‘𝐴))
3		lltr 27870	. . . 4 ⊢ ( L ‘𝐴) <<s ( R ‘𝐴)
4		fvex 6855	. . . . 5 ⊢ ( O ‘( bday ‘𝐴)) ∈ V
5		bdayon 27760	. . . . . . 7 ⊢ ( bday ‘𝑥) ∈ On
6	5	onsuci 7791	. . . . . 6 ⊢ suc ( bday ‘𝑥) ∈ On
7	6	rgenw 3056	. . . . 5 ⊢ ∀𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥) ∈ On
8		iunon 8281	. . . . 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 693	. . . 4 ⊢ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥) ∈ On
10		lrold 27905	. . . . . 6 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴))
11	10	imaeq2i 6025	. . . . 5 ⊢ ( bday “ (( L ‘𝐴) ∪ ( R ‘𝐴))) = ( bday “ ( O ‘( bday ‘𝐴)))
12		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑦 𝐴 ∈ No
13		bdayfun 27756	. . . . . . 7 ⊢ Fun bday
14	13	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ No → Fun bday )
15		fvex 6855	. . . . . . . . . 10 ⊢ ( bday ‘𝑦) ∈ V
16	15	sucid 6409	. . . . . . . . 9 ⊢ ( bday ‘𝑦) ∈ suc ( bday ‘𝑦)
17		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ( bday ‘𝑥) = ( bday ‘𝑦))
18	17	suceqd 6392	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → suc ( bday ‘𝑥) = suc ( bday ‘𝑦))
19	18	eleq2d 2823	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (( bday ‘𝑦) ∈ suc ( bday ‘𝑥) ↔ ( bday ‘𝑦) ∈ suc ( bday ‘𝑦)))
20	19	rspcev 3578	. . . . . . . . 9 ⊢ ((𝑦 ∈ ( O ‘( bday ‘𝐴)) ∧ ( bday ‘𝑦) ∈ suc ( bday ‘𝑦)) → ∃𝑥 ∈ ( O ‘( bday ‘𝐴))( bday ‘𝑦) ∈ suc ( bday ‘𝑥))
21	16, 20	mpan2 692	. . . . . . . 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 4955	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( O ‘( bday ‘𝐴))) → ( bday ‘𝑦) ∈ ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
24	12, 14, 23	funimassd 6908	. . . . 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 27803	. . . 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 1468	. . 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 27897	. . . . 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		bdayon 27760	. . . . 5 ⊢ ( bday ‘𝐴) ∈ On
32	5, 31	onsucssi 7793	. . . 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 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 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3442   ∪ cun 3901   ⊆ wss 3903  ∪ ciun 4948   class class class wbr 5100   “ cima 5635  Oncon0 6325  suc csuc 6327  Fun wfun 6494  ‘cfv 6500  (class class class)co 7368   No csur 27619   bday cbday 27621   <<s cslts 27765   |s ccuts 27767   O cold 27831   L cleft 27833   R cright 27834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-1o 8407  df-2o 8408  df-no 27622  df-lts 27623  df-bday 27624  df-slts 27766  df-cuts 27768  df-made 27835  df-old 27836  df-left 27838  df-right 27839

This theorem is referenced by:  bdayle  27924