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Theorem sstskm 10786
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 10783 . . . 4 (𝐴 ∈ 𝑉 β†’ (tarskiMapβ€˜π΄) = ∩ {π‘₯ ∈ Tarski ∣ 𝐴 ∈ π‘₯})
2 df-rab 3407 . . . . 5 {π‘₯ ∈ Tarski ∣ 𝐴 ∈ π‘₯} = {π‘₯ ∣ (π‘₯ ∈ Tarski ∧ 𝐴 ∈ π‘₯)}
32inteqi 4915 . . . 4 ∩ {π‘₯ ∈ Tarski ∣ 𝐴 ∈ π‘₯} = ∩ {π‘₯ ∣ (π‘₯ ∈ Tarski ∧ 𝐴 ∈ π‘₯)}
41, 3eqtrdi 2789 . . 3 (𝐴 ∈ 𝑉 β†’ (tarskiMapβ€˜π΄) = ∩ {π‘₯ ∣ (π‘₯ ∈ Tarski ∧ 𝐴 ∈ π‘₯)})
54sseq2d 3980 . 2 (𝐴 ∈ 𝑉 β†’ (𝐡 βŠ† (tarskiMapβ€˜π΄) ↔ 𝐡 βŠ† ∩ {π‘₯ ∣ (π‘₯ ∈ Tarski ∧ 𝐴 ∈ π‘₯)}))
6 impexp 452 . . . 4 (((π‘₯ ∈ Tarski ∧ 𝐴 ∈ π‘₯) β†’ 𝐡 βŠ† π‘₯) ↔ (π‘₯ ∈ Tarski β†’ (𝐴 ∈ π‘₯ β†’ 𝐡 βŠ† π‘₯)))
76albii 1822 . . 3 (βˆ€π‘₯((π‘₯ ∈ Tarski ∧ 𝐴 ∈ π‘₯) β†’ 𝐡 βŠ† π‘₯) ↔ βˆ€π‘₯(π‘₯ ∈ Tarski β†’ (𝐴 ∈ π‘₯ β†’ 𝐡 βŠ† π‘₯)))
8 ssintab 4930 . . 3 (𝐡 βŠ† ∩ {π‘₯ ∣ (π‘₯ ∈ Tarski ∧ 𝐴 ∈ π‘₯)} ↔ βˆ€π‘₯((π‘₯ ∈ Tarski ∧ 𝐴 ∈ π‘₯) β†’ 𝐡 βŠ† π‘₯))
9 df-ral 3062 . . 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 397  βˆ€wal 1540   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  {crab 3406   βŠ† wss 3914  βˆ© cint 4911  β€˜cfv 6500  Tarskictsk 10692  tarskiMapctskm 10781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-groth 10767
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-tsk 10693  df-tskm 10782
This theorem is referenced by: (None)
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