Theorem meetdef 18460

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 18459	. . . 4 ⊢ (𝐾 ∈ 𝑉 → dom ∧ = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺})
5	4	eleq2d 2830	. . 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 4758	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥, 𝑦} = {𝑋, 𝑦})
10	9	eleq1d 2829	. . . 4 ⊢ (𝑥 = 𝑋 → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑦} ∈ dom 𝐺))
11		preq2 4759	. . . . 5 ⊢ (𝑦 = 𝑌 → {𝑋, 𝑦} = {𝑋, 𝑌})
12	11	eleq1d 2829	. . . 4 ⊢ (𝑦 = 𝑌 → ({𝑋, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑌} ∈ dom 𝐺))
13	10, 12	opelopabg 5557	. . 3 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑍) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺} ↔ {𝑋, 𝑌} ∈ dom 𝐺))
14	7, 8, 13	syl2anc 583	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺} ↔ {𝑋, 𝑌} ∈ dom 𝐺))
15	6, 14	bitrd 279	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∧ ↔ {𝑋, 𝑌} ∈ dom 𝐺))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2108  {cpr 4650  ⟨cop 4654  {copab 5228  dom cdm 5700  ‘cfv 6573  glbcglb 18380  meetcmee 18382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-oprab 7452  df-glb 18417  df-meet 18419

This theorem is referenced by:  meetval  18461  meetcl  18462  meetdmss  18463  meeteu  18466  clatl  18578  meetdm2  48650