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Theorem cmtfvalN 38068
Description: Value of commutes relation. (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
cmtfval.b 𝐡 = (Baseβ€˜πΎ)
cmtfval.j ∨ = (joinβ€˜πΎ)
cmtfval.m ∧ = (meetβ€˜πΎ)
cmtfval.o βŠ₯ = (ocβ€˜πΎ)
cmtfval.c 𝐢 = (cmβ€˜πΎ)
Assertion
Ref Expression
cmtfvalN (𝐾 ∈ 𝐴 β†’ 𝐢 = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ π‘₯ = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ ( βŠ₯ β€˜π‘¦))))})
Distinct variable groups:   π‘₯,𝑦,𝐡   π‘₯,𝐾,𝑦
Allowed substitution hints:   𝐴(π‘₯,𝑦)   𝐢(π‘₯,𝑦)   ∨ (π‘₯,𝑦)   ∧ (π‘₯,𝑦)   βŠ₯ (π‘₯,𝑦)

Proof of Theorem cmtfvalN
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 elex 3492 . 2 (𝐾 ∈ 𝐴 β†’ 𝐾 ∈ V)
2 cmtfval.c . . 3 𝐢 = (cmβ€˜πΎ)
3 fveq2 6888 . . . . . . . 8 (𝑝 = 𝐾 β†’ (Baseβ€˜π‘) = (Baseβ€˜πΎ))
4 cmtfval.b . . . . . . . 8 𝐡 = (Baseβ€˜πΎ)
53, 4eqtr4di 2790 . . . . . . 7 (𝑝 = 𝐾 β†’ (Baseβ€˜π‘) = 𝐡)
65eleq2d 2819 . . . . . 6 (𝑝 = 𝐾 β†’ (π‘₯ ∈ (Baseβ€˜π‘) ↔ π‘₯ ∈ 𝐡))
75eleq2d 2819 . . . . . 6 (𝑝 = 𝐾 β†’ (𝑦 ∈ (Baseβ€˜π‘) ↔ 𝑦 ∈ 𝐡))
8 fveq2 6888 . . . . . . . . 9 (𝑝 = 𝐾 β†’ (joinβ€˜π‘) = (joinβ€˜πΎ))
9 cmtfval.j . . . . . . . . 9 ∨ = (joinβ€˜πΎ)
108, 9eqtr4di 2790 . . . . . . . 8 (𝑝 = 𝐾 β†’ (joinβ€˜π‘) = ∨ )
11 fveq2 6888 . . . . . . . . . 10 (𝑝 = 𝐾 β†’ (meetβ€˜π‘) = (meetβ€˜πΎ))
12 cmtfval.m . . . . . . . . . 10 ∧ = (meetβ€˜πΎ)
1311, 12eqtr4di 2790 . . . . . . . . 9 (𝑝 = 𝐾 β†’ (meetβ€˜π‘) = ∧ )
1413oveqd 7422 . . . . . . . 8 (𝑝 = 𝐾 β†’ (π‘₯(meetβ€˜π‘)𝑦) = (π‘₯ ∧ 𝑦))
15 eqidd 2733 . . . . . . . . 9 (𝑝 = 𝐾 β†’ π‘₯ = π‘₯)
16 fveq2 6888 . . . . . . . . . . 11 (𝑝 = 𝐾 β†’ (ocβ€˜π‘) = (ocβ€˜πΎ))
17 cmtfval.o . . . . . . . . . . 11 βŠ₯ = (ocβ€˜πΎ)
1816, 17eqtr4di 2790 . . . . . . . . . 10 (𝑝 = 𝐾 β†’ (ocβ€˜π‘) = βŠ₯ )
1918fveq1d 6890 . . . . . . . . 9 (𝑝 = 𝐾 β†’ ((ocβ€˜π‘)β€˜π‘¦) = ( βŠ₯ β€˜π‘¦))
2013, 15, 19oveq123d 7426 . . . . . . . 8 (𝑝 = 𝐾 β†’ (π‘₯(meetβ€˜π‘)((ocβ€˜π‘)β€˜π‘¦)) = (π‘₯ ∧ ( βŠ₯ β€˜π‘¦)))
2110, 14, 20oveq123d 7426 . . . . . . 7 (𝑝 = 𝐾 β†’ ((π‘₯(meetβ€˜π‘)𝑦)(joinβ€˜π‘)(π‘₯(meetβ€˜π‘)((ocβ€˜π‘)β€˜π‘¦))) = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ ( βŠ₯ β€˜π‘¦))))
2221eqeq2d 2743 . . . . . 6 (𝑝 = 𝐾 β†’ (π‘₯ = ((π‘₯(meetβ€˜π‘)𝑦)(joinβ€˜π‘)(π‘₯(meetβ€˜π‘)((ocβ€˜π‘)β€˜π‘¦))) ↔ π‘₯ = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ ( βŠ₯ β€˜π‘¦)))))
236, 7, 223anbi123d 1436 . . . . 5 (𝑝 = 𝐾 β†’ ((π‘₯ ∈ (Baseβ€˜π‘) ∧ 𝑦 ∈ (Baseβ€˜π‘) ∧ π‘₯ = ((π‘₯(meetβ€˜π‘)𝑦)(joinβ€˜π‘)(π‘₯(meetβ€˜π‘)((ocβ€˜π‘)β€˜π‘¦)))) ↔ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ π‘₯ = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ ( βŠ₯ β€˜π‘¦))))))
2423opabbidv 5213 . . . 4 (𝑝 = 𝐾 β†’ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ (Baseβ€˜π‘) ∧ 𝑦 ∈ (Baseβ€˜π‘) ∧ π‘₯ = ((π‘₯(meetβ€˜π‘)𝑦)(joinβ€˜π‘)(π‘₯(meetβ€˜π‘)((ocβ€˜π‘)β€˜π‘¦))))} = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ π‘₯ = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ ( βŠ₯ β€˜π‘¦))))})
25 df-cmtN 38035 . . . 4 cm = (𝑝 ∈ V ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ (Baseβ€˜π‘) ∧ 𝑦 ∈ (Baseβ€˜π‘) ∧ π‘₯ = ((π‘₯(meetβ€˜π‘)𝑦)(joinβ€˜π‘)(π‘₯(meetβ€˜π‘)((ocβ€˜π‘)β€˜π‘¦))))})
26 df-3an 1089 . . . . . 6 ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ π‘₯ = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ ( βŠ₯ β€˜π‘¦)))) ↔ ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) ∧ π‘₯ = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ ( βŠ₯ β€˜π‘¦)))))
2726opabbii 5214 . . . . 5 {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ π‘₯ = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ ( βŠ₯ β€˜π‘¦))))} = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) ∧ π‘₯ = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ ( βŠ₯ β€˜π‘¦))))}
284fvexi 6902 . . . . . . 7 𝐡 ∈ V
2928, 28xpex 7736 . . . . . 6 (𝐡 Γ— 𝐡) ∈ V
30 opabssxp 5766 . . . . . 6 {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) ∧ π‘₯ = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ ( βŠ₯ β€˜π‘¦))))} βŠ† (𝐡 Γ— 𝐡)
3129, 30ssexi 5321 . . . . 5 {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) ∧ π‘₯ = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ ( βŠ₯ β€˜π‘¦))))} ∈ V
3227, 31eqeltri 2829 . . . 4 {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ π‘₯ = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ ( βŠ₯ β€˜π‘¦))))} ∈ V
3324, 25, 32fvmpt 6995 . . 3 (𝐾 ∈ V β†’ (cmβ€˜πΎ) = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ π‘₯ = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ ( βŠ₯ β€˜π‘¦))))})
342, 33eqtrid 2784 . 2 (𝐾 ∈ V β†’ 𝐢 = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ π‘₯ = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ ( βŠ₯ β€˜π‘¦))))})
351, 34syl 17 1 (𝐾 ∈ 𝐴 β†’ 𝐢 = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ π‘₯ = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ ( βŠ₯ β€˜π‘¦))))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3474  {copab 5209   Γ— cxp 5673  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  occoc 17201  joincjn 18260  meetcmee 18261  cmccmtN 38031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-cmtN 38035
This theorem is referenced by:  cmtvalN  38069
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