Theorem cutbday 27776

Description: The birthday of the surreal cut is equal to the minimum birthday in the gap. (Contributed by Scott Fenton, 8-Dec-2021.)

Assertion

Ref	Expression
cutbday	⊢ (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem cutbday

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cutsval 27772	. . 3 ⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) = (℩𝑦 ∈ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ( bday ‘𝑦) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)})))
2	1	eqcomd 2742	. 2 ⊢ (𝐴 <<s 𝐵 → (℩𝑦 ∈ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ( bday ‘𝑦) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)})) = (𝐴 |s 𝐵))
3		cutcuts 27773	. . . 4 ⊢ (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
4		sneq 4577	. . . . . . . 8 ⊢ (𝑥 = (𝐴 |s 𝐵) → {𝑥} = {(𝐴 |s 𝐵)})
5	4	breq2d 5097	. . . . . . 7 ⊢ (𝑥 = (𝐴 |s 𝐵) → (𝐴 <<s {𝑥} ↔ 𝐴 <<s {(𝐴 |s 𝐵)}))
6	4	breq1d 5095	. . . . . . 7 ⊢ (𝑥 = (𝐴 |s 𝐵) → ({𝑥} <<s 𝐵 ↔ {(𝐴 |s 𝐵)} <<s 𝐵))
7	5, 6	anbi12d 633	. . . . . 6 ⊢ (𝑥 = (𝐴 |s 𝐵) → ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) ↔ (𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵)))
8	7	elrab 3634	. . . . 5 ⊢ ((𝐴 |s 𝐵) ∈ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ↔ ((𝐴 |s 𝐵) ∈ No ∧ (𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵)))
9		3anass 1095	. . . . 5 ⊢ (((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵) ↔ ((𝐴 |s 𝐵) ∈ No ∧ (𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵)))
10	8, 9	bitr4i 278	. . . 4 ⊢ ((𝐴 |s 𝐵) ∈ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ↔ ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
11	3, 10	sylibr 234	. . 3 ⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)})
12		conway 27771	. . 3 ⊢ (𝐴 <<s 𝐵 → ∃!𝑦 ∈ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ( bday ‘𝑦) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))
13		fveqeq2 6849	. . . 4 ⊢ (𝑦 = (𝐴 |s 𝐵) → (( bday ‘𝑦) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) ↔ ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)})))
14	13	riota2 7349	. . 3 ⊢ (((𝐴 |s 𝐵) ∈ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ∧ ∃!𝑦 ∈ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ( bday ‘𝑦) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)})) → (( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) ↔ (℩𝑦 ∈ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ( bday ‘𝑦) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)})) = (𝐴 |s 𝐵)))
15	11, 12, 14	syl2anc 585	. 2 ⊢ (𝐴 <<s 𝐵 → (( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) ↔ (℩𝑦 ∈ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ( bday ‘𝑦) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)})) = (𝐴 |s 𝐵)))
16	2, 15	mpbird 257	1 ⊢ (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃!wreu 3340  {crab 3389  {csn 4567  ∩ cint 4889   class class class wbr 5085   “ cima 5634  ‘cfv 6498  ℩crio 7323  (class class class)co 7367   No csur 27603   bday cbday 27605   <<s cslts 27749   |s ccuts 27751

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1o 8405  df-2o 8406  df-no 27606  df-lts 27607  df-bday 27608  df-slts 27750  df-cuts 27752

This theorem is referenced by:  cutsun12  27782  cutbdaybnd  27787  cutbdaybnd2  27788  cutbdaylt  27790  bday0  27803  bday1  27806  sltsbday  27909  cofcut1  27912  cofcutr  27916  oniso  28263  bdayn0p1  28361