Theorem joinval 18432

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 18429	. . . . . 6 ⊢ (𝐾 ∈ 𝑉 → ∨ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))})
5	1, 4	syl 18	. . . . 5 ⊢ (𝜑 → ∨ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))})
6	5	oveqd 7429	. . . 4 ⊢ (𝜑 → (𝑋 ∨ 𝑌) = (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌))
7	6	adantr 485	. . 3 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 ∨ 𝑌) = (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌))
8		simpr 489	. . . 4 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → {𝑋, 𝑌} ∈ dom 𝑈)
9		eqidd 2764	. . . 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 6898	. . . . . 6 ⊢ (𝜑 → (𝑈‘{𝑋, 𝑌}) ∈ V)
13		preq12 4702	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑥, 𝑦} = {𝑋, 𝑌})
14	13	eleq1d 2848	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ {𝑋, 𝑌} ∈ dom 𝑈))
15	14	3adant3 1150	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ∧ 𝑧 = (𝑈‘{𝑋, 𝑌})) → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ {𝑋, 𝑌} ∈ dom 𝑈))
16		simp3 1156	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ∧ 𝑧 = (𝑈‘{𝑋, 𝑌})) → 𝑧 = (𝑈‘{𝑋, 𝑌}))
17	13	fveq2d 6887	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑈‘{𝑥, 𝑦}) = (𝑈‘{𝑋, 𝑌}))
18	17	3adant3 1150	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ∧ 𝑧 = (𝑈‘{𝑋, 𝑌})) → (𝑈‘{𝑥, 𝑦}) = (𝑈‘{𝑋, 𝑌}))
19	16, 18	eqeq12d 2779	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ∧ 𝑧 = (𝑈‘{𝑋, 𝑌})) → (𝑧 = (𝑈‘{𝑥, 𝑦}) ↔ (𝑈‘{𝑋, 𝑌}) = (𝑈‘{𝑋, 𝑌})))
20	15, 19	anbi12d 643	. . . . . . 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 3671	. . . . . . . 8 ⊢ ∃*𝑧 𝑧 = (𝑈‘{𝑥, 𝑦})
22	21	moani 2581	. . . . . . 7 ⊢ ∃*𝑧({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))
23		eqid 2763	. . . . . . 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 7557	. . . . . 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 1398	. . . . 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 485	. . . 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 711	. . 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 2798	. 2 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 ∨ 𝑌) = (𝑈‘{𝑋, 𝑌}))
29	2, 3, 1, 10, 11	joindef 18431	. . . . . 6 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∨ ↔ {𝑋, 𝑌} ∈ dom 𝑈))
30	29	notbid 321	. . . . 5 ⊢ (𝜑 → (¬ ⟨𝑋, 𝑌⟩ ∈ dom ∨ ↔ ¬ {𝑋, 𝑌} ∈ dom 𝑈))
31		df-ov 7415	. . . . . 6 ⊢ (𝑋 ∨ 𝑌) = ( ∨ ‘⟨𝑋, 𝑌⟩)
32		ndmfv 6915	. . . . . 6 ⊢ (¬ ⟨𝑋, 𝑌⟩ ∈ dom ∨ → ( ∨ ‘⟨𝑋, 𝑌⟩) = ∅)
33	31, 32	eqtrid 2810	. . . . 5 ⊢ (¬ ⟨𝑋, 𝑌⟩ ∈ dom ∨ → (𝑋 ∨ 𝑌) = ∅)
34	30, 33	biimtrrdi 257	. . . 4 ⊢ (𝜑 → (¬ {𝑋, 𝑌} ∈ dom 𝑈 → (𝑋 ∨ 𝑌) = ∅))
35	34	imp 411	. . 3 ⊢ ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 ∨ 𝑌) = ∅)
36		ndmfv 6915	. . . 4 ⊢ (¬ {𝑋, 𝑌} ∈ dom 𝑈 → (𝑈‘{𝑋, 𝑌}) = ∅)
37	36	adantl 486	. . 3 ⊢ ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑈‘{𝑋, 𝑌}) = ∅)
38	35, 37	eqtr4d 2801	. 2 ⊢ ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 ∨ 𝑌) = (𝑈‘{𝑋, 𝑌}))
39	28, 38	pm2.61dan 824	1 ⊢ (𝜑 → (𝑋 ∨ 𝑌) = (𝑈‘{𝑋, 𝑌}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ∅c0 4287  {cpr 4592  ⟨cop 4596  dom cdm 5663  ‘cfv 6538  (class class class)co 7412  {coprab 7413  lubclub 18366  joincjn 18368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-lub 18401  df-join 18403

This theorem is referenced by:  joincl  18433  joinval2  18436  joincomALT  18456  lubsn  18539  posjidm  49733  toplatjoin  49763