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Theorem sstskm 9661
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 9658 . . . 4 (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
2 df-rab 2920 . . . . 5 {𝑥 ∈ Tarski ∣ 𝐴𝑥} = {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)}
32inteqi 4477 . . . 4 {𝑥 ∈ Tarski ∣ 𝐴𝑥} = {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)}
41, 3syl6eq 2671 . . 3 (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)})
54sseq2d 3631 . 2 (𝐴𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ 𝐵 {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)}))
6 impexp 462 . . . 4 (((𝑥 ∈ Tarski ∧ 𝐴𝑥) → 𝐵𝑥) ↔ (𝑥 ∈ Tarski → (𝐴𝑥𝐵𝑥)))
76albii 1746 . . 3 (∀𝑥((𝑥 ∈ Tarski ∧ 𝐴𝑥) → 𝐵𝑥) ↔ ∀𝑥(𝑥 ∈ Tarski → (𝐴𝑥𝐵𝑥)))
8 ssintab 4492 . . 3 (𝐵 {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)} ↔ ∀𝑥((𝑥 ∈ Tarski ∧ 𝐴𝑥) → 𝐵𝑥))
9 df-ral 2916 . . 3 (∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥) ↔ ∀𝑥(𝑥 ∈ Tarski → (𝐴𝑥𝐵𝑥)))
107, 8, 93bitr4i 292 . 2 (𝐵 {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴𝑥)} ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥))
115, 10syl6bb 276 1 (𝐴𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1480  wcel 1989  {cab 2607  wral 2911  {crab 2915  wss 3572   cint 4473  cfv 5886  Tarskictsk 9567  tarskiMapctskm 9656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904  ax-groth 9642
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-int 4474  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-iota 5849  df-fun 5888  df-fv 5894  df-tsk 9568  df-tskm 9657
This theorem is referenced by: (None)
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