Theorem meetfval 18395

Description: Value of meet function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) TODO: prove meetfval2 18396 first to reduce net proof size (existence part)?

Hypotheses

Ref	Expression
meetfval.u	⊢ 𝐺 = (glb‘𝐾)
meetfval.m	⊢ ∧ = (meet‘𝐾)

Assertion

Ref	Expression
meetfval	⊢ (𝐾 ∈ 𝑉 → ∧ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧})

Distinct variable groups:   𝑥,𝑦,𝑧,𝐾   𝑧,𝐺

Allowed substitution hints:   𝐺(𝑥,𝑦)   ∧ (𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem meetfval

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3480	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		meetfval.m	. . 3 ⊢ ∧ = (meet‘𝐾)
3		fvex 6888	. . . . . . 7 ⊢ (Base‘𝐾) ∈ V
4		moeq 3690	. . . . . . . 8 ⊢ ∃*𝑧 𝑧 = (𝐺‘{𝑥, 𝑦})
5	4	a1i 11	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → ∃*𝑧 𝑧 = (𝐺‘{𝑥, 𝑦}))
6		eqid 2735	. . . . . . 7 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))}
7	3, 3, 5, 6	oprabex 7973	. . . . . 6 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))} ∈ V
8	7	a1i 11	. . . . 5 ⊢ (𝐾 ∈ V → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))} ∈ V)
9		meetfval.u	. . . . . . . . . . . 12 ⊢ 𝐺 = (glb‘𝐾)
10	9	glbfun 18373	. . . . . . . . . . 11 ⊢ Fun 𝐺
11		funbrfv2b 6935	. . . . . . . . . . 11 ⊢ (Fun 𝐺 → ({𝑥, 𝑦}𝐺𝑧 ↔ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ (𝐺‘{𝑥, 𝑦}) = 𝑧)))
12	10, 11	ax-mp 5	. . . . . . . . . 10 ⊢ ({𝑥, 𝑦}𝐺𝑧 ↔ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ (𝐺‘{𝑥, 𝑦}) = 𝑧))
13		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐾) = (Base‘𝐾)
14		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (le‘𝐾) = (le‘𝐾)
15		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ V ∧ {𝑥, 𝑦} ∈ dom 𝐺) → 𝐾 ∈ V)
16		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ V ∧ {𝑥, 𝑦} ∈ dom 𝐺) → {𝑥, 𝑦} ∈ dom 𝐺)
17	13, 14, 9, 15, 16	glbelss 18375	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ V ∧ {𝑥, 𝑦} ∈ dom 𝐺) → {𝑥, 𝑦} ⊆ (Base‘𝐾))
18	17	ex 412	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ V → ({𝑥, 𝑦} ∈ dom 𝐺 → {𝑥, 𝑦} ⊆ (Base‘𝐾)))
19		vex 3463	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
20		vex 3463	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
21	19, 20	prss 4796	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐾))
22	18, 21	imbitrrdi 252	. . . . . . . . . . 11 ⊢ (𝐾 ∈ V → ({𝑥, 𝑦} ∈ dom 𝐺 → (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))))
23		eqcom 2742	. . . . . . . . . . . 12 ⊢ ((𝐺‘{𝑥, 𝑦}) = 𝑧 ↔ 𝑧 = (𝐺‘{𝑥, 𝑦}))
24	23	biimpi 216	. . . . . . . . . . 11 ⊢ ((𝐺‘{𝑥, 𝑦}) = 𝑧 → 𝑧 = (𝐺‘{𝑥, 𝑦}))
25	22, 24	anim12d1 610	. . . . . . . . . 10 ⊢ (𝐾 ∈ V → (({𝑥, 𝑦} ∈ dom 𝐺 ∧ (𝐺‘{𝑥, 𝑦}) = 𝑧) → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))))
26	12, 25	biimtrid 242	. . . . . . . . 9 ⊢ (𝐾 ∈ V → ({𝑥, 𝑦}𝐺𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))))
27	26	alrimiv 1927	. . . . . . . 8 ⊢ (𝐾 ∈ V → ∀𝑧({𝑥, 𝑦}𝐺𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))))
28	27	alrimiv 1927	. . . . . . 7 ⊢ (𝐾 ∈ V → ∀𝑦∀𝑧({𝑥, 𝑦}𝐺𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))))
29	28	alrimiv 1927	. . . . . 6 ⊢ (𝐾 ∈ V → ∀𝑥∀𝑦∀𝑧({𝑥, 𝑦}𝐺𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))))
30		ssoprab2 7473	. . . . . 6 ⊢ (∀𝑥∀𝑦∀𝑧({𝑥, 𝑦}𝐺𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))})
31	29, 30	syl 17	. . . . 5 ⊢ (𝐾 ∈ V → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))})
32	8, 31	ssexd 5294	. . . 4 ⊢ (𝐾 ∈ V → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧} ∈ V)
33		fveq2 6875	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = (glb‘𝐾))
34	33, 9	eqtr4di 2788	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = 𝐺)
35	34	breqd 5130	. . . . . 6 ⊢ (𝑝 = 𝐾 → ({𝑥, 𝑦} (glb‘𝑝)𝑧 ↔ {𝑥, 𝑦}𝐺𝑧))
36	35	oprabbidv 7471	. . . . 5 ⊢ (𝑝 = 𝐾 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘𝑝)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧})
37		df-meet 18357	. . . . 5 ⊢ meet = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘𝑝)𝑧})
38	36, 37	fvmptg 6983	. . . 4 ⊢ ((𝐾 ∈ V ∧ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧} ∈ V) → (meet‘𝐾) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧})
39	32, 38	mpdan 687	. . 3 ⊢ (𝐾 ∈ V → (meet‘𝐾) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧})
40	2, 39	eqtrid 2782	. 2 ⊢ (𝐾 ∈ V → ∧ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧})
41	1, 40	syl 17	1 ⊢ (𝐾 ∈ 𝑉 → ∧ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2108  ∃*wmo 2537  Vcvv 3459   ⊆ wss 3926  {cpr 4603   class class class wbr 5119  dom cdm 5654  Fun wfun 6524  ‘cfv 6530  {coprab 7404  Basecbs 17226  lecple 17276  glbcglb 18320  meetcmee 18322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-oprab 7407  df-glb 18355  df-meet 18357

This theorem is referenced by:  meetfval2  18396  meet0  18414  odujoin  18416  odumeet  18418