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Theorem tskmval 10782
Description: Value of our tarski map. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
tskmval (𝐴 ∈ 𝑉 β†’ (tarskiMapβ€˜π΄) = ∩ {π‘₯ ∈ Tarski ∣ 𝐴 ∈ π‘₯})
Distinct variable group:   π‘₯,𝐴
Allowed substitution hint:   𝑉(π‘₯)

Proof of Theorem tskmval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elex 3466 . 2 (𝐴 ∈ 𝑉 β†’ 𝐴 ∈ V)
2 grothtsk 10778 . . . . 5 βˆͺ Tarski = V
31, 2eleqtrrdi 2849 . . . 4 (𝐴 ∈ 𝑉 β†’ 𝐴 ∈ βˆͺ Tarski)
4 eluni2 4874 . . . 4 (𝐴 ∈ βˆͺ Tarski ↔ βˆƒπ‘₯ ∈ Tarski 𝐴 ∈ π‘₯)
53, 4sylib 217 . . 3 (𝐴 ∈ 𝑉 β†’ βˆƒπ‘₯ ∈ Tarski 𝐴 ∈ π‘₯)
6 intexrab 5302 . . 3 (βˆƒπ‘₯ ∈ Tarski 𝐴 ∈ π‘₯ ↔ ∩ {π‘₯ ∈ Tarski ∣ 𝐴 ∈ π‘₯} ∈ V)
75, 6sylib 217 . 2 (𝐴 ∈ 𝑉 β†’ ∩ {π‘₯ ∈ Tarski ∣ 𝐴 ∈ π‘₯} ∈ V)
8 eleq1 2826 . . . . 5 (𝑦 = 𝐴 β†’ (𝑦 ∈ π‘₯ ↔ 𝐴 ∈ π‘₯))
98rabbidv 3418 . . . 4 (𝑦 = 𝐴 β†’ {π‘₯ ∈ Tarski ∣ 𝑦 ∈ π‘₯} = {π‘₯ ∈ Tarski ∣ 𝐴 ∈ π‘₯})
109inteqd 4917 . . 3 (𝑦 = 𝐴 β†’ ∩ {π‘₯ ∈ Tarski ∣ 𝑦 ∈ π‘₯} = ∩ {π‘₯ ∈ Tarski ∣ 𝐴 ∈ π‘₯})
11 df-tskm 10781 . . 3 tarskiMap = (𝑦 ∈ V ↦ ∩ {π‘₯ ∈ Tarski ∣ 𝑦 ∈ π‘₯})
1210, 11fvmptg 6951 . 2 ((𝐴 ∈ V ∧ ∩ {π‘₯ ∈ Tarski ∣ 𝐴 ∈ π‘₯} ∈ V) β†’ (tarskiMapβ€˜π΄) = ∩ {π‘₯ ∈ Tarski ∣ 𝐴 ∈ π‘₯})
131, 7, 12syl2anc 585 1 (𝐴 ∈ 𝑉 β†’ (tarskiMapβ€˜π΄) = ∩ {π‘₯ ∈ Tarski ∣ 𝐴 ∈ π‘₯})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3074  {crab 3410  Vcvv 3448  βˆͺ cuni 4870  βˆ© cint 4912  β€˜cfv 6501  Tarskictsk 10691  tarskiMapctskm 10780
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-groth 10766
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-tsk 10692  df-tskm 10781
This theorem is referenced by:  tskmid  10783  tskmcl  10784  sstskm  10785  eltskm  10786
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