Theorem leftval 33637

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

Assertion

Ref	Expression
leftval	⊢ (𝐴 ∈ No → ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴})

Distinct variable group:   𝑥,𝐴

Proof of Theorem leftval

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2fveq3 6668	. . 3 ⊢ (𝑦 = 𝐴 → ( O ‘( bday ‘𝑦)) = ( O ‘( bday ‘𝐴)))
2		breq2 5040	. . 3 ⊢ (𝑦 = 𝐴 → (𝑥 <s 𝑦 ↔ 𝑥 <s 𝐴))
3	1, 2	rabeqbidv 3398	. 2 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ ( O ‘( bday ‘𝑦)) ∣ 𝑥 <s 𝑦} = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴})
4		df-left 33628	. 2 ⊢ L = (𝑦 ∈ No ↦ {𝑥 ∈ ( O ‘( bday ‘𝑦)) ∣ 𝑥 <s 𝑦})
5		fvex 6676	. . 3 ⊢ ( O ‘( bday ‘𝐴)) ∈ V
6	5	rabex 5206	. 2 ⊢ {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} ∈ V
7	3, 4, 6	fvmpt 6764	1 ⊢ (𝐴 ∈ No → ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴})

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {crab 3074   class class class wbr 5036  ‘cfv 6340   No csur 33440   <s cslt 33441   bday cbday 33442   O cold 33621   L cleft 33623

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6299  df-fun 6342  df-fv 6348  df-left 33628

This theorem is referenced by:  ssltleft  33644  leftssold  33652  lrold  33668  madebdaylemlrcut  33670