Theorem meetfval 18336

Description: Value of meet function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) TODO: prove meetfval2 18337 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 3492	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		meetfval.m	. . 3 ⊢ ∧ = (meet‘𝐾)
3		fvex 6901	. . . . . . 7 ⊢ (Base‘𝐾) ∈ V
4		moeq 3702	. . . . . . . 8 ⊢ ∃*𝑧 𝑧 = (𝐺‘{𝑥, 𝑦})
5	4	a1i 11	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → ∃*𝑧 𝑧 = (𝐺‘{𝑥, 𝑦}))
6		eqid 2732	. . . . . . 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 7959	. . . . . 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 18314	. . . . . . . . . . 11 ⊢ Fun 𝐺
11		funbrfv2b 6946	. . . . . . . . . . 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 2732	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐾) = (Base‘𝐾)
14		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (le‘𝐾) = (le‘𝐾)
15		simpl 483	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ V ∧ {𝑥, 𝑦} ∈ dom 𝐺) → 𝐾 ∈ V)
16		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ V ∧ {𝑥, 𝑦} ∈ dom 𝐺) → {𝑥, 𝑦} ∈ dom 𝐺)
17	13, 14, 9, 15, 16	glbelss 18316	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ V ∧ {𝑥, 𝑦} ∈ dom 𝐺) → {𝑥, 𝑦} ⊆ (Base‘𝐾))
18	17	ex 413	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ V → ({𝑥, 𝑦} ∈ dom 𝐺 → {𝑥, 𝑦} ⊆ (Base‘𝐾)))
19		vex 3478	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
20		vex 3478	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
21	19, 20	prss 4822	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐾))
22	18, 21	syl6ibr 251	. . . . . . . . . . 11 ⊢ (𝐾 ∈ V → ({𝑥, 𝑦} ∈ dom 𝐺 → (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))))
23		eqcom 2739	. . . . . . . . . . . 12 ⊢ ((𝐺‘{𝑥, 𝑦}) = 𝑧 ↔ 𝑧 = (𝐺‘{𝑥, 𝑦}))
24	23	biimpi 215	. . . . . . . . . . 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 241	. . . . . . . . 9 ⊢ (𝐾 ∈ V → ({𝑥, 𝑦}𝐺𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))))
27	26	alrimiv 1930	. . . . . . . 8 ⊢ (𝐾 ∈ V → ∀𝑧({𝑥, 𝑦}𝐺𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))))
28	27	alrimiv 1930	. . . . . . 7 ⊢ (𝐾 ∈ V → ∀𝑦∀𝑧({𝑥, 𝑦}𝐺𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))))
29	28	alrimiv 1930	. . . . . 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 5323	. . . 4 ⊢ (𝐾 ∈ V → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧} ∈ V)
33		fveq2 6888	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = (glb‘𝐾))
34	33, 9	eqtr4di 2790	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = 𝐺)
35	34	breqd 5158	. . . . . 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 18298	. . . . 5 ⊢ meet = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘𝑝)𝑧})
38	36, 37	fvmptg 6993	. . . 4 ⊢ ((𝐾 ∈ V ∧ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧} ∈ V) → (meet‘𝐾) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧})
39	32, 38	mpdan 685	. . 3 ⊢ (𝐾 ∈ V → (meet‘𝐾) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧})
40	2, 39	eqtrid 2784	. 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 205   ∧ wa 396  ∀wal 1539   = wceq 1541   ∈ wcel 2106  ∃*wmo 2532  Vcvv 3474   ⊆ wss 3947  {cpr 4629   class class class wbr 5147  dom cdm 5675  Fun wfun 6534  ‘cfv 6540  {coprab 7406  Basecbs 17140  lecple 17200  glbcglb 18259  meetcmee 18261

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-oprab 7409  df-glb 18296  df-meet 18298

This theorem is referenced by:  meetfval2  18337  meet0  18355  odujoin  18357  odumeet  18359