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Theorem cmtfvalN 36454
Description: Value of commutes relation. (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
cmtfval.b 𝐵 = (Base‘𝐾)
cmtfval.j = (join‘𝐾)
cmtfval.m = (meet‘𝐾)
cmtfval.o = (oc‘𝐾)
cmtfval.c 𝐶 = (cm‘𝐾)
Assertion
Ref Expression
cmtfvalN (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐶(𝑥,𝑦)   (𝑥,𝑦)   (𝑥,𝑦)   (𝑥,𝑦)

Proof of Theorem cmtfvalN
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 elex 3498 . 2 (𝐾𝐴𝐾 ∈ V)
2 cmtfval.c . . 3 𝐶 = (cm‘𝐾)
3 fveq2 6661 . . . . . . . 8 (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4 cmtfval.b . . . . . . . 8 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2877 . . . . . . 7 (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
65eleq2d 2901 . . . . . 6 (𝑝 = 𝐾 → (𝑥 ∈ (Base‘𝑝) ↔ 𝑥𝐵))
75eleq2d 2901 . . . . . 6 (𝑝 = 𝐾 → (𝑦 ∈ (Base‘𝑝) ↔ 𝑦𝐵))
8 fveq2 6661 . . . . . . . . 9 (𝑝 = 𝐾 → (join‘𝑝) = (join‘𝐾))
9 cmtfval.j . . . . . . . . 9 = (join‘𝐾)
108, 9syl6eqr 2877 . . . . . . . 8 (𝑝 = 𝐾 → (join‘𝑝) = )
11 fveq2 6661 . . . . . . . . . 10 (𝑝 = 𝐾 → (meet‘𝑝) = (meet‘𝐾))
12 cmtfval.m . . . . . . . . . 10 = (meet‘𝐾)
1311, 12syl6eqr 2877 . . . . . . . . 9 (𝑝 = 𝐾 → (meet‘𝑝) = )
1413oveqd 7166 . . . . . . . 8 (𝑝 = 𝐾 → (𝑥(meet‘𝑝)𝑦) = (𝑥 𝑦))
15 eqidd 2825 . . . . . . . . 9 (𝑝 = 𝐾𝑥 = 𝑥)
16 fveq2 6661 . . . . . . . . . . 11 (𝑝 = 𝐾 → (oc‘𝑝) = (oc‘𝐾))
17 cmtfval.o . . . . . . . . . . 11 = (oc‘𝐾)
1816, 17syl6eqr 2877 . . . . . . . . . 10 (𝑝 = 𝐾 → (oc‘𝑝) = )
1918fveq1d 6663 . . . . . . . . 9 (𝑝 = 𝐾 → ((oc‘𝑝)‘𝑦) = ( 𝑦))
2013, 15, 19oveq123d 7170 . . . . . . . 8 (𝑝 = 𝐾 → (𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦)) = (𝑥 ( 𝑦)))
2110, 14, 20oveq123d 7170 . . . . . . 7 (𝑝 = 𝐾 → ((𝑥(meet‘𝑝)𝑦)(join‘𝑝)(𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦))) = ((𝑥 𝑦) (𝑥 ( 𝑦))))
2221eqeq2d 2835 . . . . . 6 (𝑝 = 𝐾 → (𝑥 = ((𝑥(meet‘𝑝)𝑦)(join‘𝑝)(𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦))) ↔ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦)))))
236, 7, 223anbi123d 1433 . . . . 5 (𝑝 = 𝐾 → ((𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝) ∧ 𝑥 = ((𝑥(meet‘𝑝)𝑦)(join‘𝑝)(𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦)))) ↔ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))))
2423opabbidv 5118 . . . 4 (𝑝 = 𝐾 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝) ∧ 𝑥 = ((𝑥(meet‘𝑝)𝑦)(join‘𝑝)(𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦))))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
25 df-cmtN 36421 . . . 4 cm = (𝑝 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝) ∧ 𝑥 = ((𝑥(meet‘𝑝)𝑦)(join‘𝑝)(𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦))))})
26 df-3an 1086 . . . . . 6 ((𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦)))) ↔ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦)))))
2726opabbii 5119 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}
284fvexi 6675 . . . . . . 7 𝐵 ∈ V
2928, 28xpex 7470 . . . . . 6 (𝐵 × 𝐵) ∈ V
30 opabssxp 5630 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} ⊆ (𝐵 × 𝐵)
3129, 30ssexi 5212 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} ∈ V
3227, 31eqeltri 2912 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} ∈ V
3324, 25, 32fvmpt 6759 . . 3 (𝐾 ∈ V → (cm‘𝐾) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
342, 33syl5eq 2871 . 2 (𝐾 ∈ V → 𝐶 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
351, 34syl 17 1 (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  Vcvv 3480  {copab 5114   × cxp 5540  cfv 6343  (class class class)co 7149  Basecbs 16483  occoc 16573  joincjn 17554  meetcmee 17555  cmccmtN 36417
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7152  df-cmtN 36421
This theorem is referenced by:  cmtvalN  36455
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