Metamath Proof Explorer


Theorem mppsval

Description: Definition of a provable pre-statement, essentially just a reorganization of the arguments of df-mcls . (Contributed by Mario Carneiro, 18-Jul-2016)

Ref Expression
Hypotheses mppsval.p 𝑃 = ( mPreSt ‘ 𝑇 )
mppsval.j 𝐽 = ( mPPSt ‘ 𝑇 )
mppsval.c 𝐶 = ( mCls ‘ 𝑇 )
Assertion mppsval 𝐽 = { ⟨ ⟨ 𝑑 , ⟩ , 𝑎 ⟩ ∣ ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ 𝑃𝑎 ∈ ( 𝑑 𝐶 ) ) }

Proof

Step Hyp Ref Expression
1 mppsval.p 𝑃 = ( mPreSt ‘ 𝑇 )
2 mppsval.j 𝐽 = ( mPPSt ‘ 𝑇 )
3 mppsval.c 𝐶 = ( mCls ‘ 𝑇 )
4 fveq2 ( 𝑡 = 𝑇 → ( mPreSt ‘ 𝑡 ) = ( mPreSt ‘ 𝑇 ) )
5 4 1 eqtr4di ( 𝑡 = 𝑇 → ( mPreSt ‘ 𝑡 ) = 𝑃 )
6 5 eleq2d ( 𝑡 = 𝑇 → ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ ( mPreSt ‘ 𝑡 ) ↔ ⟨ 𝑑 , , 𝑎 ⟩ ∈ 𝑃 ) )
7 fveq2 ( 𝑡 = 𝑇 → ( mCls ‘ 𝑡 ) = ( mCls ‘ 𝑇 ) )
8 7 3 eqtr4di ( 𝑡 = 𝑇 → ( mCls ‘ 𝑡 ) = 𝐶 )
9 8 oveqd ( 𝑡 = 𝑇 → ( 𝑑 ( mCls ‘ 𝑡 ) ) = ( 𝑑 𝐶 ) )
10 9 eleq2d ( 𝑡 = 𝑇 → ( 𝑎 ∈ ( 𝑑 ( mCls ‘ 𝑡 ) ) ↔ 𝑎 ∈ ( 𝑑 𝐶 ) ) )
11 6 10 anbi12d ( 𝑡 = 𝑇 → ( ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ ( mPreSt ‘ 𝑡 ) ∧ 𝑎 ∈ ( 𝑑 ( mCls ‘ 𝑡 ) ) ) ↔ ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ 𝑃𝑎 ∈ ( 𝑑 𝐶 ) ) ) )
12 11 oprabbidv ( 𝑡 = 𝑇 → { ⟨ ⟨ 𝑑 , ⟩ , 𝑎 ⟩ ∣ ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ ( mPreSt ‘ 𝑡 ) ∧ 𝑎 ∈ ( 𝑑 ( mCls ‘ 𝑡 ) ) ) } = { ⟨ ⟨ 𝑑 , ⟩ , 𝑎 ⟩ ∣ ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ 𝑃𝑎 ∈ ( 𝑑 𝐶 ) ) } )
13 df-mpps mPPSt = ( 𝑡 ∈ V ↦ { ⟨ ⟨ 𝑑 , ⟩ , 𝑎 ⟩ ∣ ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ ( mPreSt ‘ 𝑡 ) ∧ 𝑎 ∈ ( 𝑑 ( mCls ‘ 𝑡 ) ) ) } )
14 1 fvexi 𝑃 ∈ V
15 1 2 3 mppspstlem { ⟨ ⟨ 𝑑 , ⟩ , 𝑎 ⟩ ∣ ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ 𝑃𝑎 ∈ ( 𝑑 𝐶 ) ) } ⊆ 𝑃
16 14 15 ssexi { ⟨ ⟨ 𝑑 , ⟩ , 𝑎 ⟩ ∣ ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ 𝑃𝑎 ∈ ( 𝑑 𝐶 ) ) } ∈ V
17 12 13 16 fvmpt ( 𝑇 ∈ V → ( mPPSt ‘ 𝑇 ) = { ⟨ ⟨ 𝑑 , ⟩ , 𝑎 ⟩ ∣ ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ 𝑃𝑎 ∈ ( 𝑑 𝐶 ) ) } )
18 fvprc ( ¬ 𝑇 ∈ V → ( mPPSt ‘ 𝑇 ) = ∅ )
19 df-oprab { ⟨ ⟨ 𝑑 , ⟩ , 𝑎 ⟩ ∣ ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ 𝑃𝑎 ∈ ( 𝑑 𝐶 ) ) } = { 𝑥 ∣ ∃ 𝑑𝑎 ( 𝑥 = ⟨ ⟨ 𝑑 , ⟩ , 𝑎 ⟩ ∧ ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ 𝑃𝑎 ∈ ( 𝑑 𝐶 ) ) ) }
20 abn0 ( { 𝑥 ∣ ∃ 𝑑𝑎 ( 𝑥 = ⟨ ⟨ 𝑑 , ⟩ , 𝑎 ⟩ ∧ ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ 𝑃𝑎 ∈ ( 𝑑 𝐶 ) ) ) } ≠ ∅ ↔ ∃ 𝑥𝑑𝑎 ( 𝑥 = ⟨ ⟨ 𝑑 , ⟩ , 𝑎 ⟩ ∧ ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ 𝑃𝑎 ∈ ( 𝑑 𝐶 ) ) ) )
21 elfvex ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ ( mPreSt ‘ 𝑇 ) → 𝑇 ∈ V )
22 21 1 eleq2s ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ 𝑃𝑇 ∈ V )
23 22 ad2antrl ( ( 𝑥 = ⟨ ⟨ 𝑑 , ⟩ , 𝑎 ⟩ ∧ ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ 𝑃𝑎 ∈ ( 𝑑 𝐶 ) ) ) → 𝑇 ∈ V )
24 23 exlimivv ( ∃ 𝑎 ( 𝑥 = ⟨ ⟨ 𝑑 , ⟩ , 𝑎 ⟩ ∧ ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ 𝑃𝑎 ∈ ( 𝑑 𝐶 ) ) ) → 𝑇 ∈ V )
25 24 exlimivv ( ∃ 𝑥𝑑𝑎 ( 𝑥 = ⟨ ⟨ 𝑑 , ⟩ , 𝑎 ⟩ ∧ ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ 𝑃𝑎 ∈ ( 𝑑 𝐶 ) ) ) → 𝑇 ∈ V )
26 20 25 sylbi ( { 𝑥 ∣ ∃ 𝑑𝑎 ( 𝑥 = ⟨ ⟨ 𝑑 , ⟩ , 𝑎 ⟩ ∧ ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ 𝑃𝑎 ∈ ( 𝑑 𝐶 ) ) ) } ≠ ∅ → 𝑇 ∈ V )
27 26 necon1bi ( ¬ 𝑇 ∈ V → { 𝑥 ∣ ∃ 𝑑𝑎 ( 𝑥 = ⟨ ⟨ 𝑑 , ⟩ , 𝑎 ⟩ ∧ ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ 𝑃𝑎 ∈ ( 𝑑 𝐶 ) ) ) } = ∅ )
28 19 27 syl5eq ( ¬ 𝑇 ∈ V → { ⟨ ⟨ 𝑑 , ⟩ , 𝑎 ⟩ ∣ ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ 𝑃𝑎 ∈ ( 𝑑 𝐶 ) ) } = ∅ )
29 18 28 eqtr4d ( ¬ 𝑇 ∈ V → ( mPPSt ‘ 𝑇 ) = { ⟨ ⟨ 𝑑 , ⟩ , 𝑎 ⟩ ∣ ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ 𝑃𝑎 ∈ ( 𝑑 𝐶 ) ) } )
30 17 29 pm2.61i ( mPPSt ‘ 𝑇 ) = { ⟨ ⟨ 𝑑 , ⟩ , 𝑎 ⟩ ∣ ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ 𝑃𝑎 ∈ ( 𝑑 𝐶 ) ) }
31 2 30 eqtri 𝐽 = { ⟨ ⟨ 𝑑 , ⟩ , 𝑎 ⟩ ∣ ( ⟨ 𝑑 , , 𝑎 ⟩ ∈ 𝑃𝑎 ∈ ( 𝑑 𝐶 ) ) }