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Theorem eltskm 10881
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 10877 . . 3 (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
21eleq2d 2825 . 2 (𝐴𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ 𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥}))
3 elex 3499 . . . 4 (𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥} → 𝐵 ∈ V)
43a1i 11 . . 3 (𝐴𝑉 → (𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥} → 𝐵 ∈ V))
5 tskmid 10878 . . . . 5 (𝐴𝑉𝐴 ∈ (tarskiMap‘𝐴))
6 tskmcl 10879 . . . . . 6 (tarskiMap‘𝐴) ∈ Tarski
7 eleq2 2828 . . . . . . . 8 (𝑥 = (tarskiMap‘𝐴) → (𝐴𝑥𝐴 ∈ (tarskiMap‘𝐴)))
8 eleq2 2828 . . . . . . . 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 3499 . . . 4 (𝐵 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ V)
1412, 13syl6 35 . . 3 (𝐴𝑉 → (∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥) → 𝐵 ∈ V))
15 elintrabg 4966 . . . 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 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478   cint 4951  cfv 6563  Tarskictsk 10786  tarskiMapctskm 10875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-groth 10861
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-er 8744  df-en 8985  df-dom 8986  df-tsk 10787  df-tskm 10876
This theorem is referenced by: (None)
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