Theorem meetdef 17622

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 17621	. . . 4 ⊢ (𝐾 ∈ 𝑉 → dom ∧ = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺})
5	4	eleq2d 2898	. . 3 ⊢ (𝐾 ∈ 𝑉 → (⟨𝑋, 𝑌⟩ ∈ dom ∧ ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺}))
6	1, 5	syl 17	. 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 4663	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥, 𝑦} = {𝑋, 𝑦})
10	9	eleq1d 2897	. . . 4 ⊢ (𝑥 = 𝑋 → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑦} ∈ dom 𝐺))
11		preq2 4664	. . . . 5 ⊢ (𝑦 = 𝑌 → {𝑋, 𝑦} = {𝑋, 𝑌})
12	11	eleq1d 2897	. . . 4 ⊢ (𝑦 = 𝑌 → ({𝑋, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑌} ∈ dom 𝐺))
13	10, 12	opelopabg 5418	. . 3 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑍) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺} ↔ {𝑋, 𝑌} ∈ dom 𝐺))
14	7, 8, 13	syl2anc 586	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺} ↔ {𝑋, 𝑌} ∈ dom 𝐺))
15	6, 14	bitrd 281	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∧ ↔ {𝑋, 𝑌} ∈ dom 𝐺))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1533   ∈ wcel 2110  {cpr 4563  ⟨cop 4567  {copab 5121  dom cdm 5550  ‘cfv 6350  glbcglb 17547  meetcmee 17549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-oprab 7154  df-glb 17579  df-meet 17581

This theorem is referenced by:  meetval  17623  meetcl  17624  meetdmss  17625  meeteu  17628  clatl  17720