Theorem cutbday 27854

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 27850	. . 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 2767	. 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 27851	. . . 4 ⊢ (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
4		sneq 4591	. . . . . . . 8 ⊢ (𝑥 = (𝐴 |s 𝐵) → {𝑥} = {(𝐴 |s 𝐵)})
5	4	breq2d 5111	. . . . . . 7 ⊢ (𝑥 = (𝐴 |s 𝐵) → (𝐴 <<s {𝑥} ↔ 𝐴 <<s {(𝐴 |s 𝐵)}))
6	4	breq1d 5109	. . . . . . 7 ⊢ (𝑥 = (𝐴 |s 𝐵) → ({𝑥} <<s 𝐵 ↔ {(𝐴 |s 𝐵)} <<s 𝐵))
7	5, 6	anbi12d 641	. . . . . 6 ⊢ (𝑥 = (𝐴 |s 𝐵) → ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) ↔ (𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵)))
8	7	elrab 3650	. . . . 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 1105	. . . . 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 280	. . . 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 236	. . 3 ⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)})
12		conway 27849	. . 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 6872	. . . 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 7374	. . 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 593	. 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 259	1 ⊢ (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∃!wreu 3364  {crab 3413  {csn 4581  ∩ cint 4904   class class class wbr 5099   “ cima 5648  ‘cfv 6517  ℩crio 7348  (class class class)co 7392   No csur 27681   bday cbday 27683   <<s cslts 27827   |s ccuts 27829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  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 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4905  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-ord 6345  df-on 6346  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1o 8432  df-2o 8433  df-no 27684  df-lts 27685  df-bday 27686  df-slts 27828  df-cuts 27830

This theorem is referenced by:  cutsun12  27860  cutbdaybnd  27865  cutbdaybnd2  27866  cutbdaylt  27868  bday0  27881  bday1  27884  sltsbday  27987  cofcut1  27990  cofcutr  27994  oniso  28341  bdayn0p1  28439