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Theorem meetdef 17225
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 (𝜑𝐾𝑉)
meetdef.x (𝜑𝑋𝑊)
meetdef.y (𝜑𝑌𝑍)
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.
StepHypRef Expression
1 meetdef.k . . 3 (𝜑𝐾𝑉)
2 meetdef.u . . . . 5 𝐺 = (glb‘𝐾)
3 meetdef.m . . . . 5 = (meet‘𝐾)
42, 3meetdm 17224 . . . 4 (𝐾𝑉 → dom = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺})
54eleq2d 2835 . . 3 (𝐾𝑉 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺}))
61, 5syl 17 . 2 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺}))
7 meetdef.x . . 3 (𝜑𝑋𝑊)
8 meetdef.y . . 3 (𝜑𝑌𝑍)
9 preq1 4402 . . . . 5 (𝑥 = 𝑋 → {𝑥, 𝑦} = {𝑋, 𝑦})
109eleq1d 2834 . . . 4 (𝑥 = 𝑋 → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑦} ∈ dom 𝐺))
11 preq2 4403 . . . . 5 (𝑦 = 𝑌 → {𝑋, 𝑦} = {𝑋, 𝑌})
1211eleq1d 2834 . . . 4 (𝑦 = 𝑌 → ({𝑋, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑌} ∈ dom 𝐺))
1310, 12opelopabg 5126 . . 3 ((𝑋𝑊𝑌𝑍) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺} ↔ {𝑋, 𝑌} ∈ dom 𝐺))
147, 8, 13syl2anc 565 . 2 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺} ↔ {𝑋, 𝑌} ∈ dom 𝐺))
156, 14bitrd 268 1 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ {𝑋, 𝑌} ∈ dom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1630  wcel 2144  {cpr 4316  cop 4320  {copab 4844  dom cdm 5249  cfv 6031  glbcglb 17150  meetcmee 17152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-oprab 6796  df-glb 17182  df-meet 17184
This theorem is referenced by:  meetval  17226  meetcl  17227  meetdmss  17228  meeteu  17231  clatl  17323
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