Theorem joinval 18332

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

Hypotheses

Ref	Expression
joindef.u	⊢ 𝑈 = (lub‘𝐾)
joindef.j	⊢ ∨ = (join‘𝐾)
joindef.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
joindef.x	⊢ (𝜑 → 𝑋 ∈ 𝑊)
joindef.y	⊢ (𝜑 → 𝑌 ∈ 𝑍)

Assertion

Ref	Expression
joinval	⊢ (𝜑 → (𝑋 ∨ 𝑌) = (𝑈‘{𝑋, 𝑌}))

Proof of Theorem joinval

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

Step	Hyp	Ref	Expression
1		joindef.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
2		joindef.u	. . . . . . 7 ⊢ 𝑈 = (lub‘𝐾)
3		joindef.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
4	2, 3	joinfval2 18329	. . . . . 6 ⊢ (𝐾 ∈ 𝑉 → ∨ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))})
5	1, 4	syl 17	. . . . 5 ⊢ (𝜑 → ∨ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))})
6	5	oveqd 7428	. . . 4 ⊢ (𝜑 → (𝑋 ∨ 𝑌) = (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌))
7	6	adantr 481	. . 3 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 ∨ 𝑌) = (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌))
8		simpr 485	. . . 4 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → {𝑋, 𝑌} ∈ dom 𝑈)
9		eqidd 2733	. . . 4 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑈‘{𝑋, 𝑌}) = (𝑈‘{𝑋, 𝑌}))
10		joindef.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
11		joindef.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑍)
12		fvexd 6906	. . . . . 6 ⊢ (𝜑 → (𝑈‘{𝑋, 𝑌}) ∈ V)
13		preq12 4739	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑥, 𝑦} = {𝑋, 𝑌})
14	13	eleq1d 2818	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ {𝑋, 𝑌} ∈ dom 𝑈))
15	14	3adant3 1132	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ∧ 𝑧 = (𝑈‘{𝑋, 𝑌})) → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ {𝑋, 𝑌} ∈ dom 𝑈))
16		simp3 1138	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ∧ 𝑧 = (𝑈‘{𝑋, 𝑌})) → 𝑧 = (𝑈‘{𝑋, 𝑌}))
17	13	fveq2d 6895	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑈‘{𝑥, 𝑦}) = (𝑈‘{𝑋, 𝑌}))
18	17	3adant3 1132	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ∧ 𝑧 = (𝑈‘{𝑋, 𝑌})) → (𝑈‘{𝑥, 𝑦}) = (𝑈‘{𝑋, 𝑌}))
19	16, 18	eqeq12d 2748	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ∧ 𝑧 = (𝑈‘{𝑋, 𝑌})) → (𝑧 = (𝑈‘{𝑥, 𝑦}) ↔ (𝑈‘{𝑋, 𝑌}) = (𝑈‘{𝑋, 𝑌})))
20	15, 19	anbi12d 631	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ∧ 𝑧 = (𝑈‘{𝑋, 𝑌})) → (({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦})) ↔ ({𝑋, 𝑌} ∈ dom 𝑈 ∧ (𝑈‘{𝑋, 𝑌}) = (𝑈‘{𝑋, 𝑌}))))
21		moeq 3703	. . . . . . . 8 ⊢ ∃*𝑧 𝑧 = (𝑈‘{𝑥, 𝑦})
22	21	moani 2547	. . . . . . 7 ⊢ ∃*𝑧({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))
23		eqid 2732	. . . . . . 7 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))}
24	20, 22, 23	ovigg 7555	. . . . . 6 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑍 ∧ (𝑈‘{𝑋, 𝑌}) ∈ V) → (({𝑋, 𝑌} ∈ dom 𝑈 ∧ (𝑈‘{𝑋, 𝑌}) = (𝑈‘{𝑋, 𝑌})) → (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌) = (𝑈‘{𝑋, 𝑌})))
25	10, 11, 12, 24	syl3anc 1371	. . . . 5 ⊢ (𝜑 → (({𝑋, 𝑌} ∈ dom 𝑈 ∧ (𝑈‘{𝑋, 𝑌}) = (𝑈‘{𝑋, 𝑌})) → (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌) = (𝑈‘{𝑋, 𝑌})))
26	25	adantr 481	. . . 4 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (({𝑋, 𝑌} ∈ dom 𝑈 ∧ (𝑈‘{𝑋, 𝑌}) = (𝑈‘{𝑋, 𝑌})) → (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌) = (𝑈‘{𝑋, 𝑌})))
27	8, 9, 26	mp2and 697	. . 3 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌) = (𝑈‘{𝑋, 𝑌}))
28	7, 27	eqtrd 2772	. 2 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 ∨ 𝑌) = (𝑈‘{𝑋, 𝑌}))
29	2, 3, 1, 10, 11	joindef 18331	. . . . . 6 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∨ ↔ {𝑋, 𝑌} ∈ dom 𝑈))
30	29	notbid 317	. . . . 5 ⊢ (𝜑 → (¬ ⟨𝑋, 𝑌⟩ ∈ dom ∨ ↔ ¬ {𝑋, 𝑌} ∈ dom 𝑈))
31		df-ov 7414	. . . . . 6 ⊢ (𝑋 ∨ 𝑌) = ( ∨ ‘⟨𝑋, 𝑌⟩)
32		ndmfv 6926	. . . . . 6 ⊢ (¬ ⟨𝑋, 𝑌⟩ ∈ dom ∨ → ( ∨ ‘⟨𝑋, 𝑌⟩) = ∅)
33	31, 32	eqtrid 2784	. . . . 5 ⊢ (¬ ⟨𝑋, 𝑌⟩ ∈ dom ∨ → (𝑋 ∨ 𝑌) = ∅)
34	30, 33	syl6bir 253	. . . 4 ⊢ (𝜑 → (¬ {𝑋, 𝑌} ∈ dom 𝑈 → (𝑋 ∨ 𝑌) = ∅))
35	34	imp 407	. . 3 ⊢ ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 ∨ 𝑌) = ∅)
36		ndmfv 6926	. . . 4 ⊢ (¬ {𝑋, 𝑌} ∈ dom 𝑈 → (𝑈‘{𝑋, 𝑌}) = ∅)
37	36	adantl 482	. . 3 ⊢ ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑈‘{𝑋, 𝑌}) = ∅)
38	35, 37	eqtr4d 2775	. 2 ⊢ ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 ∨ 𝑌) = (𝑈‘{𝑋, 𝑌}))
39	28, 38	pm2.61dan 811	1 ⊢ (𝜑 → (𝑋 ∨ 𝑌) = (𝑈‘{𝑋, 𝑌}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3474  ∅c0 4322  {cpr 4630  ⟨cop 4634  dom cdm 5676  ‘cfv 6543  (class class class)co 7411  {coprab 7412  lubclub 18264  joincjn 18266

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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-lub 18301  df-join 18303

This theorem is referenced by:  joincl  18333  joinval2  18336  joincomALT  18356  lubsn  18437  posjidm  47683  toplatjoin  47705