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Theorem cmtfvalN 37672
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 3463 . 2 (𝐾𝐴𝐾 ∈ V)
2 cmtfval.c . . 3 𝐶 = (cm‘𝐾)
3 fveq2 6842 . . . . . . . 8 (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4 cmtfval.b . . . . . . . 8 𝐵 = (Base‘𝐾)
53, 4eqtr4di 2794 . . . . . . 7 (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
65eleq2d 2823 . . . . . 6 (𝑝 = 𝐾 → (𝑥 ∈ (Base‘𝑝) ↔ 𝑥𝐵))
75eleq2d 2823 . . . . . 6 (𝑝 = 𝐾 → (𝑦 ∈ (Base‘𝑝) ↔ 𝑦𝐵))
8 fveq2 6842 . . . . . . . . 9 (𝑝 = 𝐾 → (join‘𝑝) = (join‘𝐾))
9 cmtfval.j . . . . . . . . 9 = (join‘𝐾)
108, 9eqtr4di 2794 . . . . . . . 8 (𝑝 = 𝐾 → (join‘𝑝) = )
11 fveq2 6842 . . . . . . . . . 10 (𝑝 = 𝐾 → (meet‘𝑝) = (meet‘𝐾))
12 cmtfval.m . . . . . . . . . 10 = (meet‘𝐾)
1311, 12eqtr4di 2794 . . . . . . . . 9 (𝑝 = 𝐾 → (meet‘𝑝) = )
1413oveqd 7374 . . . . . . . 8 (𝑝 = 𝐾 → (𝑥(meet‘𝑝)𝑦) = (𝑥 𝑦))
15 eqidd 2737 . . . . . . . . 9 (𝑝 = 𝐾𝑥 = 𝑥)
16 fveq2 6842 . . . . . . . . . . 11 (𝑝 = 𝐾 → (oc‘𝑝) = (oc‘𝐾))
17 cmtfval.o . . . . . . . . . . 11 = (oc‘𝐾)
1816, 17eqtr4di 2794 . . . . . . . . . 10 (𝑝 = 𝐾 → (oc‘𝑝) = )
1918fveq1d 6844 . . . . . . . . 9 (𝑝 = 𝐾 → ((oc‘𝑝)‘𝑦) = ( 𝑦))
2013, 15, 19oveq123d 7378 . . . . . . . 8 (𝑝 = 𝐾 → (𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦)) = (𝑥 ( 𝑦)))
2110, 14, 20oveq123d 7378 . . . . . . 7 (𝑝 = 𝐾 → ((𝑥(meet‘𝑝)𝑦)(join‘𝑝)(𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦))) = ((𝑥 𝑦) (𝑥 ( 𝑦))))
2221eqeq2d 2747 . . . . . 6 (𝑝 = 𝐾 → (𝑥 = ((𝑥(meet‘𝑝)𝑦)(join‘𝑝)(𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦))) ↔ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦)))))
236, 7, 223anbi123d 1436 . . . . 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 5171 . . . 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 37639 . . . 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 1089 . . . . . 6 ((𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦)))) ↔ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦)))))
2726opabbii 5172 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}
284fvexi 6856 . . . . . . 7 𝐵 ∈ V
2928, 28xpex 7687 . . . . . 6 (𝐵 × 𝐵) ∈ V
30 opabssxp 5724 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} ⊆ (𝐵 × 𝐵)
3129, 30ssexi 5279 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} ∈ V
3227, 31eqeltri 2834 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} ∈ V
3324, 25, 32fvmpt 6948 . . 3 (𝐾 ∈ V → (cm‘𝐾) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
342, 33eqtrid 2788 . 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 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3445  {copab 5167   × cxp 5631  cfv 6496  (class class class)co 7357  Basecbs 17083  occoc 17141  joincjn 18200  meetcmee 18201  cmccmtN 37635
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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-cmtN 37639
This theorem is referenced by:  cmtvalN  37673
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