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Theorem sstskm 10880
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 10877 . . . 4 (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
2 df-rab 3434 . . . . 5 {𝑥 ∈ Tarski ∣ 𝐴𝑥} = {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)}
32inteqi 4955 . . . 4 {𝑥 ∈ Tarski ∣ 𝐴𝑥} = {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)}
41, 3eqtrdi 2791 . . 3 (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)})
54sseq2d 4028 . 2 (𝐴𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ 𝐵 {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)}))
6 impexp 450 . . . 4 (((𝑥 ∈ Tarski ∧ 𝐴𝑥) → 𝐵𝑥) ↔ (𝑥 ∈ Tarski → (𝐴𝑥𝐵𝑥)))
76albii 1816 . . 3 (∀𝑥((𝑥 ∈ Tarski ∧ 𝐴𝑥) → 𝐵𝑥) ↔ ∀𝑥(𝑥 ∈ Tarski → (𝐴𝑥𝐵𝑥)))
8 ssintab 4970 . . 3 (𝐵 {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)} ↔ ∀𝑥((𝑥 ∈ Tarski ∧ 𝐴𝑥) → 𝐵𝑥))
9 df-ral 3060 . . 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 1535  wcel 2106  {cab 2712  wral 3059  {crab 3433  wss 3963   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-pr 5438  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-iota 6516  df-fun 6565  df-fv 6571  df-tsk 10787  df-tskm 10876
This theorem is referenced by: (None)
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