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Theorem tskmval 9699
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 3243 . 2 (𝐴𝑉𝐴 ∈ V)
2 grothtsk 9695 . . . . 5 Tarski = V
31, 2syl6eleqr 2741 . . . 4 (𝐴𝑉𝐴 Tarski)
4 eluni2 4472 . . . 4 (𝐴 Tarski ↔ ∃𝑥 ∈ Tarski 𝐴𝑥)
53, 4sylib 208 . . 3 (𝐴𝑉 → ∃𝑥 ∈ Tarski 𝐴𝑥)
6 intexrab 4853 . . 3 (∃𝑥 ∈ Tarski 𝐴𝑥 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ∈ V)
75, 6sylib 208 . 2 (𝐴𝑉 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ∈ V)
8 eleq1 2718 . . . . 5 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
98rabbidv 3220 . . . 4 (𝑦 = 𝐴 → {𝑥 ∈ Tarski ∣ 𝑦𝑥} = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
109inteqd 4512 . . 3 (𝑦 = 𝐴 {𝑥 ∈ Tarski ∣ 𝑦𝑥} = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
11 df-tskm 9698 . . 3 tarskiMap = (𝑦 ∈ V ↦ {𝑥 ∈ Tarski ∣ 𝑦𝑥})
1210, 11fvmptg 6319 . 2 ((𝐴 ∈ V ∧ {𝑥 ∈ Tarski ∣ 𝐴𝑥} ∈ V) → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
131, 7, 12syl2anc 694 1 (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  wrex 2942  {crab 2945  Vcvv 3231   cuni 4468   cint 4507  cfv 5926  Tarskictsk 9608  tarskiMapctskm 9697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-groth 9683
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-tsk 9609  df-tskm 9698
This theorem is referenced by:  tskmid  9700  tskmcl  9701  sstskm  9702  eltskm  9703
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