Theorem meetval 18312

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 18309	. . . . . 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 7375	. . . 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 2737	. . . 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 6849	. . . . . 6 ⊢ (𝜑 → (𝐺‘{𝑋, 𝑌}) ∈ V)
13		preq12 4692	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑥, 𝑦} = {𝑋, 𝑌})
14	13	eleq1d 2821	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑌} ∈ dom 𝐺))
15	14	3adant3 1132	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ∧ 𝑧 = (𝐺‘{𝑋, 𝑌})) → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑌} ∈ dom 𝐺))
16		simp3 1138	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ∧ 𝑧 = (𝐺‘{𝑋, 𝑌})) → 𝑧 = (𝐺‘{𝑋, 𝑌}))
17	13	fveq2d 6838	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐺‘{𝑥, 𝑦}) = (𝐺‘{𝑋, 𝑌}))
18	17	3adant3 1132	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ∧ 𝑧 = (𝐺‘{𝑋, 𝑌})) → (𝐺‘{𝑥, 𝑦}) = (𝐺‘{𝑋, 𝑌}))
19	16, 18	eqeq12d 2752	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ∧ 𝑧 = (𝐺‘{𝑋, 𝑌})) → (𝑧 = (𝐺‘{𝑥, 𝑦}) ↔ (𝐺‘{𝑋, 𝑌}) = (𝐺‘{𝑋, 𝑌})))
20	15, 19	anbi12d 632	. . . . . . 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 3665	. . . . . . . 8 ⊢ ∃*𝑧 𝑧 = (𝐺‘{𝑥, 𝑦})
22	21	moani 2553	. . . . . . 7 ⊢ ∃*𝑧({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))
23		eqid 2736	. . . . . . 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 7503	. . . . . 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 1373	. . . . 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 699	. . 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 2771	. 2 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝐺) → (𝑋 ∧ 𝑌) = (𝐺‘{𝑋, 𝑌}))
29	2, 3, 1, 10, 11	meetdef 18311	. . . . . 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 7361	. . . . . 6 ⊢ (𝑋 ∧ 𝑌) = ( ∧ ‘⟨𝑋, 𝑌⟩)
32		ndmfv 6866	. . . . . 6 ⊢ (¬ ⟨𝑋, 𝑌⟩ ∈ dom ∧ → ( ∧ ‘⟨𝑋, 𝑌⟩) = ∅)
33	31, 32	eqtrid 2783	. . . . 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 6866	. . . 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 2774	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  Vcvv 3440  ∅c0 4285  {cpr 4582  ⟨cop 4586  dom cdm 5624  ‘cfv 6492  (class class class)co 7358  {coprab 7359  glbcglb 18233  meetcmee 18235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-glb 18268  df-meet 18270

This theorem is referenced by:  meetcl  18313  meetval2  18316  meetcomALT  18324  pmapmeet  40029  diameetN  41312  dihmeetlem2N  41555  dihmeetcN  41558  dihmeet  41599  posmidm  49214  toplatmeet  49244