Theorem joindef 18554

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 18553	. . . 4 ⊢ (𝐾 ∈ 𝑉 → dom ∨ = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈})
5	4	eleq2d 2846	. . 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 4690	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥, 𝑦} = {𝑋, 𝑦})
10	9	eleq1d 2845	. . . 4 ⊢ (𝑥 = 𝑋 → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ {𝑋, 𝑦} ∈ dom 𝑈))
11		preq2 4691	. . . . 5 ⊢ (𝑦 = 𝑌 → {𝑋, 𝑦} = {𝑋, 𝑌})
12	11	eleq1d 2845	. . . 4 ⊢ (𝑦 = 𝑌 → ({𝑋, 𝑦} ∈ dom 𝑈 ↔ {𝑋, 𝑌} ∈ dom 𝑈))
13	10, 12	opelopabg 5512	. . 3 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑍) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈} ↔ {𝑋, 𝑌} ∈ dom 𝑈))
14	7, 8, 13	syl2anc 596	. 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2144  {cpr 4586  ⟨cop 4834  {copab 5166  dom cdm 5650  ‘cfv 6539  lubclub 18489  joincjn 18491

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-oprab 7425  df-lub 18524  df-join 18526

This theorem is used by:  joinval  18555  joincl  18556  joindmss  18557  joineu  18560  clatl  18688  joindm2  50176