Theorem meetval 18324

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 18321	. . . . . 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 7385	. . . 4 ⊢ (𝜑 → (𝑋 ∧ 𝑌) = (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))}𝑌))
7	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝐺) → (𝑋 ∧ 𝑌) = (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))}𝑌))
8		simpr 484	. . . 4 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝐺) → {𝑋, 𝑌} ∈ dom 𝐺)
9		eqidd 2738	. . . 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 6857	. . . . . 6 ⊢ (𝜑 → (𝐺‘{𝑋, 𝑌}) ∈ V)
13		preq12 4694	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑥, 𝑦} = {𝑋, 𝑌})
14	13	eleq1d 2822	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑌} ∈ dom 𝐺))
15	14	3adant3 1133	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ∧ 𝑧 = (𝐺‘{𝑋, 𝑌})) → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑌} ∈ dom 𝐺))
16		simp3 1139	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ∧ 𝑧 = (𝐺‘{𝑋, 𝑌})) → 𝑧 = (𝐺‘{𝑋, 𝑌}))
17	13	fveq2d 6846	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐺‘{𝑥, 𝑦}) = (𝐺‘{𝑋, 𝑌}))
18	17	3adant3 1133	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ∧ 𝑧 = (𝐺‘{𝑋, 𝑌})) → (𝐺‘{𝑥, 𝑦}) = (𝐺‘{𝑋, 𝑌}))
19	16, 18	eqeq12d 2753	. . . . . . . 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 3667	. . . . . . . 8 ⊢ ∃*𝑧 𝑧 = (𝐺‘{𝑥, 𝑦})
22	21	moani 2554	. . . . . . 7 ⊢ ∃*𝑧({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))
23		eqid 2737	. . . . . . 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 7513	. . . . . 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 1374	. . . . 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 480	. . . 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 700	. . 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 2772	. 2 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝐺) → (𝑋 ∧ 𝑌) = (𝐺‘{𝑋, 𝑌}))
29	2, 3, 1, 10, 11	meetdef 18323	. . . . . 6 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∧ ↔ {𝑋, 𝑌} ∈ dom 𝐺))
30	29	notbid 318	. . . . 5 ⊢ (𝜑 → (¬ ⟨𝑋, 𝑌⟩ ∈ dom ∧ ↔ ¬ {𝑋, 𝑌} ∈ dom 𝐺))
31		df-ov 7371	. . . . . 6 ⊢ (𝑋 ∧ 𝑌) = ( ∧ ‘⟨𝑋, 𝑌⟩)
32		ndmfv 6874	. . . . . 6 ⊢ (¬ ⟨𝑋, 𝑌⟩ ∈ dom ∧ → ( ∧ ‘⟨𝑋, 𝑌⟩) = ∅)
33	31, 32	eqtrid 2784	. . . . 5 ⊢ (¬ ⟨𝑋, 𝑌⟩ ∈ dom ∧ → (𝑋 ∧ 𝑌) = ∅)
34	30, 33	biimtrrdi 254	. . . 4 ⊢ (𝜑 → (¬ {𝑋, 𝑌} ∈ dom 𝐺 → (𝑋 ∧ 𝑌) = ∅))
35	34	imp 406	. . 3 ⊢ ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝐺) → (𝑋 ∧ 𝑌) = ∅)
36		ndmfv 6874	. . . 4 ⊢ (¬ {𝑋, 𝑌} ∈ dom 𝐺 → (𝐺‘{𝑋, 𝑌}) = ∅)
37	36	adantl 481	. . 3 ⊢ ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝐺) → (𝐺‘{𝑋, 𝑌}) = ∅)
38	35, 37	eqtr4d 2775	. 2 ⊢ ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝐺) → (𝑋 ∧ 𝑌) = (𝐺‘{𝑋, 𝑌}))
39	28, 38	pm2.61dan 813	1 ⊢ (𝜑 → (𝑋 ∧ 𝑌) = (𝐺‘{𝑋, 𝑌}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ∅c0 4287  {cpr 4584  ⟨cop 4588  dom cdm 5632  ‘cfv 6500  (class class class)co 7368  {coprab 7369  glbcglb 18245  meetcmee 18247

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 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-glb 18280  df-meet 18282

This theorem is referenced by:  meetcl  18325  meetval2  18328  meetcomALT  18336  pmapmeet  40143  diameetN  41426  dihmeetlem2N  41669  dihmeetcN  41672  dihmeet  41713  posmidm  49326  toplatmeet  49356