Theorem meetfval 18352

Description: Value of meet function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) TODO: prove meetfval2 18353 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 3487	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		meetfval.m	. . 3 ⊢ ∧ = (meet‘𝐾)
3		fvex 6898	. . . . . . 7 ⊢ (Base‘𝐾) ∈ V
4		moeq 3698	. . . . . . . 8 ⊢ ∃*𝑧 𝑧 = (𝐺‘{𝑥, 𝑦})
5	4	a1i 11	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → ∃*𝑧 𝑧 = (𝐺‘{𝑥, 𝑦}))
6		eqid 2726	. . . . . . 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 7962	. . . . . 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 18330	. . . . . . . . . . 11 ⊢ Fun 𝐺
11		funbrfv2b 6943	. . . . . . . . . . 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 2726	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐾) = (Base‘𝐾)
14		eqid 2726	. . . . . . . . . . . . . 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 18332	. . . . . . . . . . . . 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 3472	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
20		vex 3472	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
21	19, 20	prss 4818	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐾))
22	18, 21	imbitrrdi 251	. . . . . . . . . . 11 ⊢ (𝐾 ∈ V → ({𝑥, 𝑦} ∈ dom 𝐺 → (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))))
23		eqcom 2733	. . . . . . . . . . . 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 609	. . . . . . . . . 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 1922	. . . . . . . 8 ⊢ (𝐾 ∈ V → ∀𝑧({𝑥, 𝑦}𝐺𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))))
28	27	alrimiv 1922	. . . . . . 7 ⊢ (𝐾 ∈ V → ∀𝑦∀𝑧({𝑥, 𝑦}𝐺𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))))
29	28	alrimiv 1922	. . . . . 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 5317	. . . 4 ⊢ (𝐾 ∈ V → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧} ∈ V)
33		fveq2 6885	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = (glb‘𝐾))
34	33, 9	eqtr4di 2784	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = 𝐺)
35	34	breqd 5152	. . . . . 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 18314	. . . . 5 ⊢ meet = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘𝑝)𝑧})
38	36, 37	fvmptg 6990	. . . 4 ⊢ ((𝐾 ∈ V ∧ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧} ∈ V) → (meet‘𝐾) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧})
39	32, 38	mpdan 684	. . 3 ⊢ (𝐾 ∈ V → (meet‘𝐾) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧})
40	2, 39	eqtrid 2778	. 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 395  ∀wal 1531   = wceq 1533   ∈ wcel 2098  ∃*wmo 2526  Vcvv 3468   ⊆ wss 3943  {cpr 4625   class class class wbr 5141  dom cdm 5669  Fun wfun 6531  ‘cfv 6537  {coprab 7406  Basecbs 17153  lecple 17213  glbcglb 18275  meetcmee 18277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-oprab 7409  df-glb 18312  df-meet 18314

This theorem is referenced by:  meetfval2  18353  meet0  18371  odujoin  18373  odumeet  18375