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Theorem sstskm 10252
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 10249 . . . 4 (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
2 df-rab 3144 . . . . 5 {𝑥 ∈ Tarski ∣ 𝐴𝑥} = {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)}
32inteqi 4871 . . . 4 {𝑥 ∈ Tarski ∣ 𝐴𝑥} = {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)}
41, 3syl6eq 2869 . . 3 (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)})
54sseq2d 3996 . 2 (𝐴𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ 𝐵 {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)}))
6 impexp 451 . . . 4 (((𝑥 ∈ Tarski ∧ 𝐴𝑥) → 𝐵𝑥) ↔ (𝑥 ∈ Tarski → (𝐴𝑥𝐵𝑥)))
76albii 1811 . . 3 (∀𝑥((𝑥 ∈ Tarski ∧ 𝐴𝑥) → 𝐵𝑥) ↔ ∀𝑥(𝑥 ∈ Tarski → (𝐴𝑥𝐵𝑥)))
8 ssintab 4884 . . 3 (𝐵 {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)} ↔ ∀𝑥((𝑥 ∈ Tarski ∧ 𝐴𝑥) → 𝐵𝑥))
9 df-ral 3140 . . 3 (∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥) ↔ ∀𝑥(𝑥 ∈ Tarski → (𝐴𝑥𝐵𝑥)))
107, 8, 93bitr4i 304 . 2 (𝐵 {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)} ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥))
115, 10syl6bb 288 1 (𝐴𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526  wcel 2105  {cab 2796  wral 3135  {crab 3139  wss 3933   cint 4867  cfv 6348  Tarskictsk 10158  tarskiMapctskm 10247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-groth 10233
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-tsk 10159  df-tskm 10248
This theorem is referenced by: (None)
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