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Theorem sstskm 10860
Description: Being a part of (tarskiMapβ€˜π΄). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
sstskm (𝐴 ∈ 𝑉 β†’ (𝐡 βŠ† (tarskiMapβ€˜π΄) ↔ βˆ€π‘₯ ∈ Tarski (𝐴 ∈ π‘₯ β†’ 𝐡 βŠ† π‘₯)))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐡
Allowed substitution hint:   𝑉(π‘₯)

Proof of Theorem sstskm
StepHypRef Expression
1 tskmval 10857 . . . 4 (𝐴 ∈ 𝑉 β†’ (tarskiMapβ€˜π΄) = ∩ {π‘₯ ∈ Tarski ∣ 𝐴 ∈ π‘₯})
2 df-rab 3429 . . . . 5 {π‘₯ ∈ Tarski ∣ 𝐴 ∈ π‘₯} = {π‘₯ ∣ (π‘₯ ∈ Tarski ∧ 𝐴 ∈ π‘₯)}
32inteqi 4949 . . . 4 ∩ {π‘₯ ∈ Tarski ∣ 𝐴 ∈ π‘₯} = ∩ {π‘₯ ∣ (π‘₯ ∈ Tarski ∧ 𝐴 ∈ π‘₯)}
41, 3eqtrdi 2784 . . 3 (𝐴 ∈ 𝑉 β†’ (tarskiMapβ€˜π΄) = ∩ {π‘₯ ∣ (π‘₯ ∈ Tarski ∧ 𝐴 ∈ π‘₯)})
54sseq2d 4011 . 2 (𝐴 ∈ 𝑉 β†’ (𝐡 βŠ† (tarskiMapβ€˜π΄) ↔ 𝐡 βŠ† ∩ {π‘₯ ∣ (π‘₯ ∈ Tarski ∧ 𝐴 ∈ π‘₯)}))
6 impexp 450 . . . 4 (((π‘₯ ∈ Tarski ∧ 𝐴 ∈ π‘₯) β†’ 𝐡 βŠ† π‘₯) ↔ (π‘₯ ∈ Tarski β†’ (𝐴 ∈ π‘₯ β†’ 𝐡 βŠ† π‘₯)))
76albii 1814 . . 3 (βˆ€π‘₯((π‘₯ ∈ Tarski ∧ 𝐴 ∈ π‘₯) β†’ 𝐡 βŠ† π‘₯) ↔ βˆ€π‘₯(π‘₯ ∈ Tarski β†’ (𝐴 ∈ π‘₯ β†’ 𝐡 βŠ† π‘₯)))
8 ssintab 4964 . . 3 (𝐡 βŠ† ∩ {π‘₯ ∣ (π‘₯ ∈ Tarski ∧ 𝐴 ∈ π‘₯)} ↔ βˆ€π‘₯((π‘₯ ∈ Tarski ∧ 𝐴 ∈ π‘₯) β†’ 𝐡 βŠ† π‘₯))
9 df-ral 3058 . . 3 (βˆ€π‘₯ ∈ Tarski (𝐴 ∈ π‘₯ β†’ 𝐡 βŠ† π‘₯) ↔ βˆ€π‘₯(π‘₯ ∈ Tarski β†’ (𝐴 ∈ π‘₯ β†’ 𝐡 βŠ† π‘₯)))
107, 8, 93bitr4i 303 . 2 (𝐡 βŠ† ∩ {π‘₯ ∣ (π‘₯ ∈ Tarski ∧ 𝐴 ∈ π‘₯)} ↔ βˆ€π‘₯ ∈ Tarski (𝐴 ∈ π‘₯ β†’ 𝐡 βŠ† π‘₯))
115, 10bitrdi 287 1 (𝐴 ∈ 𝑉 β†’ (𝐡 βŠ† (tarskiMapβ€˜π΄) ↔ βˆ€π‘₯ ∈ Tarski (𝐴 ∈ π‘₯ β†’ 𝐡 βŠ† π‘₯)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395  βˆ€wal 1532   ∈ wcel 2099  {cab 2705  βˆ€wral 3057  {crab 3428   βŠ† wss 3945  βˆ© cint 4945  β€˜cfv 6543  Tarskictsk 10766  tarskiMapctskm 10855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424  ax-groth 10841
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-int 4946  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-iota 6495  df-fun 6545  df-fv 6551  df-tsk 10767  df-tskm 10856
This theorem is referenced by: (None)
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