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Theorem cmtfvalN 38714
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 6902 . . . . . . . 8 (𝑝 = 𝐾 β†’ (Baseβ€˜π‘) = (Baseβ€˜πΎ))
4 cmtfval.b . . . . . . . 8 𝐡 = (Baseβ€˜πΎ)
53, 4eqtr4di 2786 . . . . . . 7 (𝑝 = 𝐾 β†’ (Baseβ€˜π‘) = 𝐡)
65eleq2d 2815 . . . . . 6 (𝑝 = 𝐾 β†’ (π‘₯ ∈ (Baseβ€˜π‘) ↔ π‘₯ ∈ 𝐡))
75eleq2d 2815 . . . . . 6 (𝑝 = 𝐾 β†’ (𝑦 ∈ (Baseβ€˜π‘) ↔ 𝑦 ∈ 𝐡))
8 fveq2 6902 . . . . . . . . 9 (𝑝 = 𝐾 β†’ (joinβ€˜π‘) = (joinβ€˜πΎ))
9 cmtfval.j . . . . . . . . 9 ∨ = (joinβ€˜πΎ)
108, 9eqtr4di 2786 . . . . . . . 8 (𝑝 = 𝐾 β†’ (joinβ€˜π‘) = ∨ )
11 fveq2 6902 . . . . . . . . . 10 (𝑝 = 𝐾 β†’ (meetβ€˜π‘) = (meetβ€˜πΎ))
12 cmtfval.m . . . . . . . . . 10 ∧ = (meetβ€˜πΎ)
1311, 12eqtr4di 2786 . . . . . . . . 9 (𝑝 = 𝐾 β†’ (meetβ€˜π‘) = ∧ )
1413oveqd 7443 . . . . . . . 8 (𝑝 = 𝐾 β†’ (π‘₯(meetβ€˜π‘)𝑦) = (π‘₯ ∧ 𝑦))
15 eqidd 2729 . . . . . . . . 9 (𝑝 = 𝐾 β†’ π‘₯ = π‘₯)
16 fveq2 6902 . . . . . . . . . . 11 (𝑝 = 𝐾 β†’ (ocβ€˜π‘) = (ocβ€˜πΎ))
17 cmtfval.o . . . . . . . . . . 11 βŠ₯ = (ocβ€˜πΎ)
1816, 17eqtr4di 2786 . . . . . . . . . 10 (𝑝 = 𝐾 β†’ (ocβ€˜π‘) = βŠ₯ )
1918fveq1d 6904 . . . . . . . . 9 (𝑝 = 𝐾 β†’ ((ocβ€˜π‘)β€˜π‘¦) = ( βŠ₯ β€˜π‘¦))
2013, 15, 19oveq123d 7447 . . . . . . . 8 (𝑝 = 𝐾 β†’ (π‘₯(meetβ€˜π‘)((ocβ€˜π‘)β€˜π‘¦)) = (π‘₯ ∧ ( βŠ₯ β€˜π‘¦)))
2110, 14, 20oveq123d 7447 . . . . . . 7 (𝑝 = 𝐾 β†’ ((π‘₯(meetβ€˜π‘)𝑦)(joinβ€˜π‘)(π‘₯(meetβ€˜π‘)((ocβ€˜π‘)β€˜π‘¦))) = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ ( βŠ₯ β€˜π‘¦))))
2221eqeq2d 2739 . . . . . 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 1432 . . . . 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 5218 . . . 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 38681 . . . 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 1086 . . . . . 6 ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ π‘₯ = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ ( βŠ₯ β€˜π‘¦)))) ↔ ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) ∧ π‘₯ = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ ( βŠ₯ β€˜π‘¦)))))
2726opabbii 5219 . . . . 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 6916 . . . . . . 7 𝐡 ∈ V
2928, 28xpex 7761 . . . . . 6 (𝐡 Γ— 𝐡) ∈ V
30 opabssxp 5774 . . . . . 6 {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) ∧ π‘₯ = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ ( βŠ₯ β€˜π‘¦))))} βŠ† (𝐡 Γ— 𝐡)
3129, 30ssexi 5326 . . . . 5 {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) ∧ π‘₯ = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ ( βŠ₯ β€˜π‘¦))))} ∈ V
3227, 31eqeltri 2825 . . . 4 {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ π‘₯ = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ ( βŠ₯ β€˜π‘¦))))} ∈ V
3324, 25, 32fvmpt 7010 . . 3 (𝐾 ∈ V β†’ (cmβ€˜πΎ) = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ π‘₯ = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ ( βŠ₯ β€˜π‘¦))))})
342, 33eqtrid 2780 . 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3473  {copab 5214   Γ— cxp 5680  β€˜cfv 6553  (class class class)co 7426  Basecbs 17187  occoc 17248  joincjn 18310  meetcmee 18311  cmccmtN 38677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-cmtN 38681
This theorem is referenced by:  cmtvalN  38715
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