Theorem leftf 33976

Description: The functionality of the left options function. (Contributed by Scott Fenton, 6-Aug-2024.)

Assertion

Ref	Expression
leftf	⊢ L : No ⟶𝒫 No

Proof of Theorem leftf

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-left 33961	. 2 ⊢ L = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥})
2		bdayelon 33898	. . . . . . . 8 ⊢ ( bday ‘𝑥) ∈ On
3		oldf 33968	. . . . . . . . 9 ⊢ O :On⟶𝒫 No
4	3	ffvelrni 6942	. . . . . . . 8 ⊢ (( bday ‘𝑥) ∈ On → ( O ‘( bday ‘𝑥)) ∈ 𝒫 No )
5	2, 4	mp1i 13	. . . . . . 7 ⊢ (𝑥 ∈ No → ( O ‘( bday ‘𝑥)) ∈ 𝒫 No )
6	5	elpwid 4541	. . . . . 6 ⊢ (𝑥 ∈ No → ( O ‘( bday ‘𝑥)) ⊆ No )
7	6	sselda 3917	. . . . 5 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ ( O ‘( bday ‘𝑥))) → 𝑦 ∈ No )
8	7	a1d 25	. . . 4 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ ( O ‘( bday ‘𝑥))) → (𝑦 <s 𝑥 → 𝑦 ∈ No ))
9	8	ralrimiva 3107	. . 3 ⊢ (𝑥 ∈ No → ∀𝑦 ∈ ( O ‘( bday ‘𝑥))(𝑦 <s 𝑥 → 𝑦 ∈ No ))
10		fvex 6769	. . . . . 6 ⊢ ( O ‘( bday ‘𝑥)) ∈ V
11	10	rabex 5251	. . . . 5 ⊢ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥} ∈ V
12	11	elpw 4534	. . . 4 ⊢ ({𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥} ∈ 𝒫 No ↔ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥} ⊆ No )
13		rabss 4001	. . . 4 ⊢ ({𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥} ⊆ No ↔ ∀𝑦 ∈ ( O ‘( bday ‘𝑥))(𝑦 <s 𝑥 → 𝑦 ∈ No ))
14	12, 13	bitri 274	. . 3 ⊢ ({𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥} ∈ 𝒫 No ↔ ∀𝑦 ∈ ( O ‘( bday ‘𝑥))(𝑦 <s 𝑥 → 𝑦 ∈ No ))
15	9, 14	sylibr 233	. 2 ⊢ (𝑥 ∈ No → {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥} ∈ 𝒫 No )
16	1, 15	fmpti 6968	1 ⊢ L : No ⟶𝒫 No

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  ∀wral 3063  {crab 3067   ⊆ wss 3883  𝒫 cpw 4530   class class class wbr 5070  Oncon0 6251  ⟶wf 6414  ‘cfv 6418   No csur 33770   <s cslt 33771   bday cbday 33772   O cold 33954   L cleft 33956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-1o 8267  df-2o 8268  df-no 33773  df-slt 33774  df-bday 33775  df-sslt 33903  df-scut 33905  df-made 33958  df-old 33959  df-left 33961

This theorem is referenced by:  ssltleft  33981  lrold  34004