Theorem meetdef 18440

Description: Two ways to say that a meet is defined. (Contributed by NM, 9-Sep-2018.)

Hypotheses

Ref	Expression
meetdef.u	⊢ 𝐺 = (glb‘𝐾)
meetdef.m	⊢ ∧ = (meet‘𝐾)
meetdef.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
meetdef.x	⊢ (𝜑 → 𝑋 ∈ 𝑊)
meetdef.y	⊢ (𝜑 → 𝑌 ∈ 𝑍)

Assertion

Ref	Expression
meetdef	⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∧ ↔ {𝑋, 𝑌} ∈ dom 𝐺))

Proof of Theorem meetdef

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

Step	Hyp	Ref	Expression
1		meetdef.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
2		meetdef.u	. . . . 5 ⊢ 𝐺 = (glb‘𝐾)
3		meetdef.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
4	2, 3	meetdm 18439	. . . 4 ⊢ (𝐾 ∈ 𝑉 → dom ∧ = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺})
5	4	eleq2d 2855	. . 3 ⊢ (𝐾 ∈ 𝑉 → (⟨𝑋, 𝑌⟩ ∈ dom ∧ ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺}))
6	1, 5	syl 18	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∧ ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺}))
7		meetdef.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
8		meetdef.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑍)
9		preq1 4701	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥, 𝑦} = {𝑋, 𝑦})
10	9	eleq1d 2854	. . . 4 ⊢ (𝑥 = 𝑋 → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑦} ∈ dom 𝐺))
11		preq2 4702	. . . . 5 ⊢ (𝑦 = 𝑌 → {𝑋, 𝑦} = {𝑋, 𝑌})
12	11	eleq1d 2854	. . . 4 ⊢ (𝑦 = 𝑌 → ({𝑋, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑌} ∈ dom 𝐺))
13	10, 12	opelopabg 5521	. . 3 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑍) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺} ↔ {𝑋, 𝑌} ∈ dom 𝐺))
14	7, 8, 13	syl2anc 595	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺} ↔ {𝑋, 𝑌} ∈ dom 𝐺))
15	6, 14	bitrd 282	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∧ ↔ {𝑋, 𝑌} ∈ dom 𝐺))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567   ∈ wcel 2149  {cpr 4593  ⟨cop 4597  {copab 5174  dom cdm 5659  ‘cfv 6533  glbcglb 18362  meetcmee 18364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7365  df-oprab 7412  df-glb 18397  df-meet 18399

This theorem is referenced by:  meetval  18441  meetcl  18442  meetdmss  18443  meeteu  18446  clatl  18560  meetdm2  49626