Theorem joindef 18375

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 18374	. . . 4 ⊢ (𝐾 ∈ 𝑉 → dom ∨ = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈})
5	4	eleq2d 2815	. . 3 ⊢ (𝐾 ∈ 𝑉 → (⟨𝑋, 𝑌⟩ ∈ dom ∨ ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈}))
6	1, 5	syl 17	. 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 4742	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥, 𝑦} = {𝑋, 𝑦})
10	9	eleq1d 2814	. . . 4 ⊢ (𝑥 = 𝑋 → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ {𝑋, 𝑦} ∈ dom 𝑈))
11		preq2 4743	. . . . 5 ⊢ (𝑦 = 𝑌 → {𝑋, 𝑦} = {𝑋, 𝑌})
12	11	eleq1d 2814	. . . 4 ⊢ (𝑦 = 𝑌 → ({𝑋, 𝑦} ∈ dom 𝑈 ↔ {𝑋, 𝑌} ∈ dom 𝑈))
13	10, 12	opelopabg 5544	. . 3 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑍) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈} ↔ {𝑋, 𝑌} ∈ dom 𝑈))
14	7, 8, 13	syl2anc 582	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈} ↔ {𝑋, 𝑌} ∈ dom 𝑈))
15	6, 14	bitrd 278	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∨ ↔ {𝑋, 𝑌} ∈ dom 𝑈))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  {cpr 4634  ⟨cop 4638  {copab 5214  dom cdm 5682  ‘cfv 6553  lubclub 18308  joincjn 18310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-oprab 7430  df-lub 18345  df-join 18347

This theorem is referenced by:  joinval  18376  joincl  18377  joindmss  18378  joineu  18381  clatl  18507  joindm2  48065