Theorem lubval 18315

Description: Value of the least upper bound function of a poset. Out-of-domain arguments (those not satisfying 𝑆 ∈ dom 𝑈) are allowed for convenience, evaluating to the empty set. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.)

Hypotheses

Ref	Expression
lubval.b	⊢ 𝐵 = (Base‘𝐾)
lubval.l	⊢ ≤ = (le‘𝐾)
lubval.u	⊢ 𝑈 = (lub‘𝐾)
lubval.p	⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
lubval.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
lubval.s	⊢ (𝜑 → 𝑆 ⊆ 𝐵)

Assertion

Ref	Expression
lubval	⊢ (𝜑 → (𝑈‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓))

Distinct variable groups:   𝑥,𝑧,𝐵   𝑥,𝑦,𝐾,𝑧   𝑥,𝑆,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)   𝐵(𝑦)   𝑈(𝑥,𝑦,𝑧)   ≤ (𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem lubval

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lubval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		lubval.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
3		lubval.u	. . . . 5 ⊢ 𝑈 = (lub‘𝐾)
4		biid 261	. . . . 5 ⊢ ((∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
5		lubval.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
6	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → 𝐾 ∈ 𝑉)
7	1, 2, 3, 4, 6	lubfval 18309	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → 𝑈 = ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))}))
8	7	fveq1d 6838	. . 3 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → (𝑈‘𝑆) = (((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))})‘𝑆))
9		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → 𝑆 ∈ dom 𝑈)
10		lubval.p	. . . . . 6 ⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
11	1, 2, 3, 10, 6, 9	lubeu 18314	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → ∃!𝑥 ∈ 𝐵 𝜓)
12		raleq 3293	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ↔ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥))
13		raleq 3293	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 ↔ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧))
14	13	imbi1d 341	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → ((∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
15	14	ralbidv 3161	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
16	12, 15	anbi12d 633	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ((∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))
17	16, 10	bitr4di 289	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ 𝜓))
18	17	reubidv 3359	. . . . 5 ⊢ (𝑠 = 𝑆 → (∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ ∃!𝑥 ∈ 𝐵 𝜓))
19	9, 11, 18	elabd 3625	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → 𝑆 ∈ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))})
20	19	fvresd 6856	. . 3 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → (((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))})‘𝑆) = ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))‘𝑆))
21		lubval.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
22	21	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → 𝑆 ⊆ 𝐵)
23	1	fvexi 6850	. . . . . 6 ⊢ 𝐵 ∈ V
24	23	elpw2 5272	. . . . 5 ⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵)
25	22, 24	sylibr 234	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → 𝑆 ∈ 𝒫 𝐵)
26	17	riotabidv 7321	. . . . 5 ⊢ (𝑠 = 𝑆 → (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) = (℩𝑥 ∈ 𝐵 𝜓))
27		eqid 2737	. . . . 5 ⊢ (𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))) = (𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))
28		riotaex 7323	. . . . 5 ⊢ (℩𝑥 ∈ 𝐵 𝜓) ∈ V
29	26, 27, 28	fvmpt 6943	. . . 4 ⊢ (𝑆 ∈ 𝒫 𝐵 → ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓))
30	25, 29	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓))
31	8, 20, 30	3eqtrd 2776	. 2 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → (𝑈‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓))
32		ndmfv 6868	. . . 4 ⊢ (¬ 𝑆 ∈ dom 𝑈 → (𝑈‘𝑆) = ∅)
33	32	adantl 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝑈) → (𝑈‘𝑆) = ∅)
34	1, 2, 3, 10, 5	lubeldm 18312	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓)))
35	34	biimprd 248	. . . . . 6 ⊢ (𝜑 → ((𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓) → 𝑆 ∈ dom 𝑈))
36	21, 35	mpand 696	. . . . 5 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 → 𝑆 ∈ dom 𝑈))
37	36	con3dimp 408	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝑈) → ¬ ∃!𝑥 ∈ 𝐵 𝜓)
38		riotaund 7358	. . . 4 ⊢ (¬ ∃!𝑥 ∈ 𝐵 𝜓 → (℩𝑥 ∈ 𝐵 𝜓) = ∅)
39	37, 38	syl 17	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝑈) → (℩𝑥 ∈ 𝐵 𝜓) = ∅)
40	33, 39	eqtr4d 2775	. 2 ⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝑈) → (𝑈‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓))
41	31, 40	pm2.61dan 813	1 ⊢ (𝜑 → (𝑈‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃!wreu 3341   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542   class class class wbr 5086   ↦ cmpt 5167  dom cdm 5626   ↾ cres 5628  ‘cfv 6494  ℩crio 7318  Basecbs 17174  lecple 17222  lubclub 18270

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 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-lub 18305

This theorem is referenced by:  lubcl  18316  lubprop  18317  lubid  18321  joinval2  18340  poslubd  18372  lubun  18476  toslub  33052  lub0N  39653  glbconN  39841