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Theorem sstskm 10882
Description: Being a part of (tarskiMap‘𝐴). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
sstskm (𝐴𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem sstskm
StepHypRef Expression
1 tskmval 10879 . . . 4 (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
2 df-rab 3437 . . . . 5 {𝑥 ∈ Tarski ∣ 𝐴𝑥} = {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)}
32inteqi 4950 . . . 4 {𝑥 ∈ Tarski ∣ 𝐴𝑥} = {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)}
41, 3eqtrdi 2793 . . 3 (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)})
54sseq2d 4016 . 2 (𝐴𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ 𝐵 {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)}))
6 impexp 450 . . . 4 (((𝑥 ∈ Tarski ∧ 𝐴𝑥) → 𝐵𝑥) ↔ (𝑥 ∈ Tarski → (𝐴𝑥𝐵𝑥)))
76albii 1819 . . 3 (∀𝑥((𝑥 ∈ Tarski ∧ 𝐴𝑥) → 𝐵𝑥) ↔ ∀𝑥(𝑥 ∈ Tarski → (𝐴𝑥𝐵𝑥)))
8 ssintab 4965 . . 3 (𝐵 {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)} ↔ ∀𝑥((𝑥 ∈ Tarski ∧ 𝐴𝑥) → 𝐵𝑥))
9 df-ral 3062 . . 3 (∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥) ↔ ∀𝑥(𝑥 ∈ Tarski → (𝐴𝑥𝐵𝑥)))
107, 8, 93bitr4i 303 . 2 (𝐵 {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)} ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥))
115, 10bitrdi 287 1 (𝐴𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538  wcel 2108  {cab 2714  wral 3061  {crab 3436  wss 3951   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-pr 5432  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-iota 6514  df-fun 6563  df-fv 6569  df-tsk 10789  df-tskm 10878
This theorem is referenced by: (None)
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