Theorem meetval 17709

Description: Meet value. Since both sides evaluate to ∅ when they don't exist, for convenience we drop the {𝑋, 𝑌} ∈ dom 𝐺 requirement. (Contributed by NM, 9-Sep-2018.)

Hypotheses

Ref	Expression
meetdef.u	⊢ 𝐺 = (glb‘𝐾)
meetdef.m	⊢ ∧ = (meet‘𝐾)
meetdef.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
meetdef.x	⊢ (𝜑 → 𝑋 ∈ 𝑊)
meetdef.y	⊢ (𝜑 → 𝑌 ∈ 𝑍)

Assertion

Ref	Expression
meetval	⊢ (𝜑 → (𝑋 ∧ 𝑌) = (𝐺‘{𝑋, 𝑌}))

Proof of Theorem meetval

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		meetdef.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
2		meetdef.u	. . . . . . 7 ⊢ 𝐺 = (glb‘𝐾)
3		meetdef.m	. . . . . . 7 ⊢ ∧ = (meet‘𝐾)
4	2, 3	meetfval2 17706	. . . . . 6 ⊢ (𝐾 ∈ 𝑉 → ∧ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))})
5	1, 4	syl 17	. . . . 5 ⊢ (𝜑 → ∧ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))})
6	5	oveqd 7173	. . . 4 ⊢ (𝜑 → (𝑋 ∧ 𝑌) = (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))}𝑌))
7	6	adantr 484	. . 3 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝐺) → (𝑋 ∧ 𝑌) = (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))}𝑌))
8		simpr 488	. . . 4 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝐺) → {𝑋, 𝑌} ∈ dom 𝐺)
9		eqidd 2759	. . . 4 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝐺) → (𝐺‘{𝑋, 𝑌}) = (𝐺‘{𝑋, 𝑌}))
10		meetdef.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
11		meetdef.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑍)
12		fvexd 6678	. . . . . 6 ⊢ (𝜑 → (𝐺‘{𝑋, 𝑌}) ∈ V)
13		preq12 4631	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑥, 𝑦} = {𝑋, 𝑌})
14	13	eleq1d 2836	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑌} ∈ dom 𝐺))
15	14	3adant3 1129	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ∧ 𝑧 = (𝐺‘{𝑋, 𝑌})) → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑌} ∈ dom 𝐺))
16		simp3 1135	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ∧ 𝑧 = (𝐺‘{𝑋, 𝑌})) → 𝑧 = (𝐺‘{𝑋, 𝑌}))
17	13	fveq2d 6667	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐺‘{𝑥, 𝑦}) = (𝐺‘{𝑋, 𝑌}))
18	17	3adant3 1129	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ∧ 𝑧 = (𝐺‘{𝑋, 𝑌})) → (𝐺‘{𝑥, 𝑦}) = (𝐺‘{𝑋, 𝑌}))
19	16, 18	eqeq12d 2774	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ∧ 𝑧 = (𝐺‘{𝑋, 𝑌})) → (𝑧 = (𝐺‘{𝑥, 𝑦}) ↔ (𝐺‘{𝑋, 𝑌}) = (𝐺‘{𝑋, 𝑌})))
20	15, 19	anbi12d 633	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ∧ 𝑧 = (𝐺‘{𝑋, 𝑌})) → (({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦})) ↔ ({𝑋, 𝑌} ∈ dom 𝐺 ∧ (𝐺‘{𝑋, 𝑌}) = (𝐺‘{𝑋, 𝑌}))))
21		moeq 3623	. . . . . . . 8 ⊢ ∃*𝑧 𝑧 = (𝐺‘{𝑥, 𝑦})
22	21	moani 2571	. . . . . . 7 ⊢ ∃*𝑧({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))
23		eqid 2758	. . . . . . 7 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))}
24	20, 22, 23	ovigg 7296	. . . . . 6 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑍 ∧ (𝐺‘{𝑋, 𝑌}) ∈ V) → (({𝑋, 𝑌} ∈ dom 𝐺 ∧ (𝐺‘{𝑋, 𝑌}) = (𝐺‘{𝑋, 𝑌})) → (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))}𝑌) = (𝐺‘{𝑋, 𝑌})))
25	10, 11, 12, 24	syl3anc 1368	. . . . 5 ⊢ (𝜑 → (({𝑋, 𝑌} ∈ dom 𝐺 ∧ (𝐺‘{𝑋, 𝑌}) = (𝐺‘{𝑋, 𝑌})) → (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))}𝑌) = (𝐺‘{𝑋, 𝑌})))
26	25	adantr 484	. . . 4 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝐺) → (({𝑋, 𝑌} ∈ dom 𝐺 ∧ (𝐺‘{𝑋, 𝑌}) = (𝐺‘{𝑋, 𝑌})) → (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))}𝑌) = (𝐺‘{𝑋, 𝑌})))
27	8, 9, 26	mp2and 698	. . 3 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝐺) → (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))}𝑌) = (𝐺‘{𝑋, 𝑌}))
28	7, 27	eqtrd 2793	. 2 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝐺) → (𝑋 ∧ 𝑌) = (𝐺‘{𝑋, 𝑌}))
29	2, 3, 1, 10, 11	meetdef 17708	. . . . . 6 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∧ ↔ {𝑋, 𝑌} ∈ dom 𝐺))
30	29	notbid 321	. . . . 5 ⊢ (𝜑 → (¬ ⟨𝑋, 𝑌⟩ ∈ dom ∧ ↔ ¬ {𝑋, 𝑌} ∈ dom 𝐺))
31		df-ov 7159	. . . . . 6 ⊢ (𝑋 ∧ 𝑌) = ( ∧ ‘⟨𝑋, 𝑌⟩)
32		ndmfv 6693	. . . . . 6 ⊢ (¬ ⟨𝑋, 𝑌⟩ ∈ dom ∧ → ( ∧ ‘⟨𝑋, 𝑌⟩) = ∅)
33	31, 32	syl5eq 2805	. . . . 5 ⊢ (¬ ⟨𝑋, 𝑌⟩ ∈ dom ∧ → (𝑋 ∧ 𝑌) = ∅)
34	30, 33	syl6bir 257	. . . 4 ⊢ (𝜑 → (¬ {𝑋, 𝑌} ∈ dom 𝐺 → (𝑋 ∧ 𝑌) = ∅))
35	34	imp 410	. . 3 ⊢ ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝐺) → (𝑋 ∧ 𝑌) = ∅)
36		ndmfv 6693	. . . 4 ⊢ (¬ {𝑋, 𝑌} ∈ dom 𝐺 → (𝐺‘{𝑋, 𝑌}) = ∅)
37	36	adantl 485	. . 3 ⊢ ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝐺) → (𝐺‘{𝑋, 𝑌}) = ∅)
38	35, 37	eqtr4d 2796	. 2 ⊢ ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝐺) → (𝑋 ∧ 𝑌) = (𝐺‘{𝑋, 𝑌}))
39	28, 38	pm2.61dan 812	1 ⊢ (𝜑 → (𝑋 ∧ 𝑌) = (𝐺‘{𝑋, 𝑌}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ∅c0 4227  {cpr 4527  ⟨cop 4531  dom cdm 5528  ‘cfv 6340  (class class class)co 7156  {coprab 7157  glbcglb 17633  meetcmee 17635

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-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465

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-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  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-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-glb 17665  df-meet 17667

This theorem is referenced by:  meetcl  17710  meetval2  17713  meetcomALT  17721  pmapmeet  37384  diameetN  38667  dihmeetlem2N  38910  dihmeetcN  38913  dihmeet  38954