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Theorem tskmval 9996
Description: Value of our tarski map. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
tskmval (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem tskmval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elex 3413 . 2 (𝐴𝑉𝐴 ∈ V)
2 grothtsk 9992 . . . . 5 Tarski = V
31, 2syl6eleqr 2869 . . . 4 (𝐴𝑉𝐴 Tarski)
4 eluni2 4675 . . . 4 (𝐴 Tarski ↔ ∃𝑥 ∈ Tarski 𝐴𝑥)
53, 4sylib 210 . . 3 (𝐴𝑉 → ∃𝑥 ∈ Tarski 𝐴𝑥)
6 intexrab 5057 . . 3 (∃𝑥 ∈ Tarski 𝐴𝑥 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ∈ V)
75, 6sylib 210 . 2 (𝐴𝑉 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ∈ V)
8 eleq1 2846 . . . . 5 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
98rabbidv 3385 . . . 4 (𝑦 = 𝐴 → {𝑥 ∈ Tarski ∣ 𝑦𝑥} = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
109inteqd 4715 . . 3 (𝑦 = 𝐴 {𝑥 ∈ Tarski ∣ 𝑦𝑥} = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
11 df-tskm 9995 . . 3 tarskiMap = (𝑦 ∈ V ↦ {𝑥 ∈ Tarski ∣ 𝑦𝑥})
1210, 11fvmptg 6540 . 2 ((𝐴 ∈ V ∧ {𝑥 ∈ Tarski ∣ 𝐴𝑥} ∈ V) → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
131, 7, 12syl2anc 579 1 (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2106  wrex 3090  {crab 3093  Vcvv 3397   cuni 4671   cint 4710  cfv 6135  Tarskictsk 9905  tarskiMapctskm 9994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-groth 9980
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-int 4711  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-tsk 9906  df-tskm 9995
This theorem is referenced by:  tskmid  9997  tskmcl  9998  sstskm  9999  eltskm  10000
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