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Theorem tskmval 10254
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 3462 . 2 (𝐴𝑉𝐴 ∈ V)
2 grothtsk 10250 . . . . 5 Tarski = V
31, 2eleqtrrdi 2904 . . . 4 (𝐴𝑉𝐴 Tarski)
4 eluni2 4807 . . . 4 (𝐴 Tarski ↔ ∃𝑥 ∈ Tarski 𝐴𝑥)
53, 4sylib 221 . . 3 (𝐴𝑉 → ∃𝑥 ∈ Tarski 𝐴𝑥)
6 intexrab 5210 . . 3 (∃𝑥 ∈ Tarski 𝐴𝑥 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ∈ V)
75, 6sylib 221 . 2 (𝐴𝑉 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ∈ V)
8 eleq1 2880 . . . . 5 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
98rabbidv 3430 . . . 4 (𝑦 = 𝐴 → {𝑥 ∈ Tarski ∣ 𝑦𝑥} = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
109inteqd 4846 . . 3 (𝑦 = 𝐴 {𝑥 ∈ Tarski ∣ 𝑦𝑥} = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
11 df-tskm 10253 . . 3 tarskiMap = (𝑦 ∈ V ↦ {𝑥 ∈ Tarski ∣ 𝑦𝑥})
1210, 11fvmptg 6747 . 2 ((𝐴 ∈ V ∧ {𝑥 ∈ Tarski ∣ 𝐴𝑥} ∈ V) → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
131, 7, 12syl2anc 587 1 (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  wrex 3110  {crab 3113  Vcvv 3444   cuni 4803   cint 4841  cfv 6328  Tarskictsk 10163  tarskiMapctskm 10252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-groth 10238
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-tsk 10164  df-tskm 10253
This theorem is referenced by:  tskmid  10255  tskmcl  10256  sstskm  10257  eltskm  10258
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