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Theorem eltskm 10883
Description: Belonging to (tarskiMap‘𝐴). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
Assertion
Ref Expression
eltskm (𝐴𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem eltskm
StepHypRef Expression
1 tskmval 10879 . . 3 (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
21eleq2d 2827 . 2 (𝐴𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ 𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥}))
3 elex 3501 . . . 4 (𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥} → 𝐵 ∈ V)
43a1i 11 . . 3 (𝐴𝑉 → (𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥} → 𝐵 ∈ V))
5 tskmid 10880 . . . . 5 (𝐴𝑉𝐴 ∈ (tarskiMap‘𝐴))
6 tskmcl 10881 . . . . . 6 (tarskiMap‘𝐴) ∈ Tarski
7 eleq2 2830 . . . . . . . 8 (𝑥 = (tarskiMap‘𝐴) → (𝐴𝑥𝐴 ∈ (tarskiMap‘𝐴)))
8 eleq2 2830 . . . . . . . 8 (𝑥 = (tarskiMap‘𝐴) → (𝐵𝑥𝐵 ∈ (tarskiMap‘𝐴)))
97, 8imbi12d 344 . . . . . . 7 (𝑥 = (tarskiMap‘𝐴) → ((𝐴𝑥𝐵𝑥) ↔ (𝐴 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ (tarskiMap‘𝐴))))
109rspcv 3618 . . . . . 6 ((tarskiMap‘𝐴) ∈ Tarski → (∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥) → (𝐴 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ (tarskiMap‘𝐴))))
116, 10ax-mp 5 . . . . 5 (∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥) → (𝐴 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ (tarskiMap‘𝐴)))
125, 11syl5com 31 . . . 4 (𝐴𝑉 → (∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥) → 𝐵 ∈ (tarskiMap‘𝐴)))
13 elex 3501 . . . 4 (𝐵 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ V)
1412, 13syl6 35 . . 3 (𝐴𝑉 → (∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥) → 𝐵 ∈ V))
15 elintrabg 4961 . . . 4 (𝐵 ∈ V → (𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥)))
1615a1i 11 . . 3 (𝐴𝑉 → (𝐵 ∈ V → (𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥))))
174, 14, 16pm5.21ndd 379 . 2 (𝐴𝑉 → (𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥)))
182, 17bitrd 279 1 (𝐴𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480   cint 4946  cfv 6561  Tarskictsk 10788  tarskiMapctskm 10877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-groth 10863
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-er 8745  df-en 8986  df-dom 8987  df-tsk 10789  df-tskm 10878
This theorem is referenced by: (None)
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