Theorem joindef 18418

Description: Two ways to say that a join is defined. (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
joindef	⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∨ ↔ {𝑋, 𝑌} ∈ dom 𝑈))

Proof of Theorem joindef

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

Step	Hyp	Ref	Expression
1		joindef.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
2		joindef.u	. . . . 5 ⊢ 𝑈 = (lub‘𝐾)
3		joindef.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
4	2, 3	joindm 18417	. . . 4 ⊢ (𝐾 ∈ 𝑉 → dom ∨ = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈})
5	4	eleq2d 2851	. . 3 ⊢ (𝐾 ∈ 𝑉 → (⟨𝑋, 𝑌⟩ ∈ dom ∨ ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈}))
6	1, 5	syl 18	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∨ ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈}))
7		joindef.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
8		joindef.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑍)
9		preq1 4695	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥, 𝑦} = {𝑋, 𝑦})
10	9	eleq1d 2850	. . . 4 ⊢ (𝑥 = 𝑋 → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ {𝑋, 𝑦} ∈ dom 𝑈))
11		preq2 4696	. . . . 5 ⊢ (𝑦 = 𝑌 → {𝑋, 𝑦} = {𝑋, 𝑌})
12	11	eleq1d 2850	. . . 4 ⊢ (𝑦 = 𝑌 → ({𝑋, 𝑦} ∈ dom 𝑈 ↔ {𝑋, 𝑌} ∈ dom 𝑈))
13	10, 12	opelopabg 5513	. . 3 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑍) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈} ↔ {𝑋, 𝑌} ∈ dom 𝑈))
14	7, 8, 13	syl2anc 595	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈} ↔ {𝑋, 𝑌} ∈ dom 𝑈))
15	6, 14	bitrd 282	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∨ ↔ {𝑋, 𝑌} ∈ dom 𝑈))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1563   ∈ wcel 2145  {cpr 4587  ⟨cop 4591  {copab 5166  dom cdm 5651  ‘cfv 6525  lubclub 18353  joincjn 18355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-oprab 7404  df-lub 18388  df-join 18390

This theorem is referenced by:  joinval  18419  joincl  18420  joindmss  18421  joineu  18424  clatl  18552  joindm2  49598