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Theorem sstskm 9543
 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 9540 . . . 4 (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
2 df-rab 2905 . . . . 5 {𝑥 ∈ Tarski ∣ 𝐴𝑥} = {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)}
32inteqi 4414 . . . 4 {𝑥 ∈ Tarski ∣ 𝐴𝑥} = {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)}
41, 3syl6eq 2660 . . 3 (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)})
54sseq2d 3596 . 2 (𝐴𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ 𝐵 {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)}))
6 impexp 461 . . . 4 (((𝑥 ∈ Tarski ∧ 𝐴𝑥) → 𝐵𝑥) ↔ (𝑥 ∈ Tarski → (𝐴𝑥𝐵𝑥)))
76albii 1737 . . 3 (∀𝑥((𝑥 ∈ Tarski ∧ 𝐴𝑥) → 𝐵𝑥) ↔ ∀𝑥(𝑥 ∈ Tarski → (𝐴𝑥𝐵𝑥)))
8 ssintab 4429 . . 3 (𝐵 {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)} ↔ ∀𝑥((𝑥 ∈ Tarski ∧ 𝐴𝑥) → 𝐵𝑥))
9 df-ral 2901 . . 3 (∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥) ↔ ∀𝑥(𝑥 ∈ Tarski → (𝐴𝑥𝐵𝑥)))
107, 8, 93bitr4i 291 . 2 (𝐵 {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)} ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥))
115, 10syl6bb 275 1 (𝐴𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   ∈ wcel 1977  {cab 2596  ∀wral 2896  {crab 2900   ⊆ wss 3540  ∩ cint 4410  ‘cfv 5804  Tarskictsk 9449  tarskiMapctskm 9538 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-groth 9524 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-tsk 9450  df-tskm 9539 This theorem is referenced by: (None)
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