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Theorem cmtfvalN 35286
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 3430 . 2 (𝐾𝐴𝐾 ∈ V)
2 cmtfval.c . . 3 𝐶 = (cm‘𝐾)
3 fveq2 6434 . . . . . . . 8 (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4 cmtfval.b . . . . . . . 8 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2880 . . . . . . 7 (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
65eleq2d 2893 . . . . . 6 (𝑝 = 𝐾 → (𝑥 ∈ (Base‘𝑝) ↔ 𝑥𝐵))
75eleq2d 2893 . . . . . 6 (𝑝 = 𝐾 → (𝑦 ∈ (Base‘𝑝) ↔ 𝑦𝐵))
8 fveq2 6434 . . . . . . . . 9 (𝑝 = 𝐾 → (join‘𝑝) = (join‘𝐾))
9 cmtfval.j . . . . . . . . 9 = (join‘𝐾)
108, 9syl6eqr 2880 . . . . . . . 8 (𝑝 = 𝐾 → (join‘𝑝) = )
11 fveq2 6434 . . . . . . . . . 10 (𝑝 = 𝐾 → (meet‘𝑝) = (meet‘𝐾))
12 cmtfval.m . . . . . . . . . 10 = (meet‘𝐾)
1311, 12syl6eqr 2880 . . . . . . . . 9 (𝑝 = 𝐾 → (meet‘𝑝) = )
1413oveqd 6923 . . . . . . . 8 (𝑝 = 𝐾 → (𝑥(meet‘𝑝)𝑦) = (𝑥 𝑦))
15 eqidd 2827 . . . . . . . . 9 (𝑝 = 𝐾𝑥 = 𝑥)
16 fveq2 6434 . . . . . . . . . . 11 (𝑝 = 𝐾 → (oc‘𝑝) = (oc‘𝐾))
17 cmtfval.o . . . . . . . . . . 11 = (oc‘𝐾)
1816, 17syl6eqr 2880 . . . . . . . . . 10 (𝑝 = 𝐾 → (oc‘𝑝) = )
1918fveq1d 6436 . . . . . . . . 9 (𝑝 = 𝐾 → ((oc‘𝑝)‘𝑦) = ( 𝑦))
2013, 15, 19oveq123d 6927 . . . . . . . 8 (𝑝 = 𝐾 → (𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦)) = (𝑥 ( 𝑦)))
2110, 14, 20oveq123d 6927 . . . . . . 7 (𝑝 = 𝐾 → ((𝑥(meet‘𝑝)𝑦)(join‘𝑝)(𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦))) = ((𝑥 𝑦) (𝑥 ( 𝑦))))
2221eqeq2d 2836 . . . . . 6 (𝑝 = 𝐾 → (𝑥 = ((𝑥(meet‘𝑝)𝑦)(join‘𝑝)(𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦))) ↔ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦)))))
236, 7, 223anbi123d 1566 . . . . 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 4940 . . . 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 35253 . . . 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 1115 . . . . . 6 ((𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦)))) ↔ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦)))))
2726opabbii 4941 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}
284fvexi 6448 . . . . . . 7 𝐵 ∈ V
2928, 28xpex 7224 . . . . . 6 (𝐵 × 𝐵) ∈ V
30 opabssxp 5429 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} ⊆ (𝐵 × 𝐵)
3129, 30ssexi 5029 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} ∈ V
3227, 31eqeltri 2903 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} ∈ V
3324, 25, 32fvmpt 6530 . . 3 (𝐾 ∈ V → (cm‘𝐾) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
342, 33syl5eq 2874 . 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 386  w3a 1113   = wceq 1658  wcel 2166  Vcvv 3415  {copab 4936   × cxp 5341  cfv 6124  (class class class)co 6906  Basecbs 16223  occoc 16314  joincjn 17298  meetcmee 17299  cmccmtN 35249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-iota 6087  df-fun 6126  df-fv 6132  df-ov 6909  df-cmtN 35253
This theorem is referenced by:  cmtvalN  35287
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