Theorem sstskm 10836

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	⊢ (𝐴 ∈ 𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem sstskm

Step	Hyp	Ref	Expression
1		tskmval 10833	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
2		df-rab 3433	. . . . 5 ⊢ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} = {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)}
3	2	inteqi 4954	. . . 4 ⊢ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} = ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)}
4	1, 3	eqtrdi 2788	. . 3 ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)})
5	4	sseq2d 4014	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ 𝐵 ⊆ ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)}))
6		impexp 451	. . . 4 ⊢ (((𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥) → 𝐵 ⊆ 𝑥) ↔ (𝑥 ∈ Tarski → (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥)))
7	6	albii 1821	. . 3 ⊢ (∀𝑥((𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥) → 𝐵 ⊆ 𝑥) ↔ ∀𝑥(𝑥 ∈ Tarski → (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥)))
8		ssintab 4969	. . 3 ⊢ (𝐵 ⊆ ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)} ↔ ∀𝑥((𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥) → 𝐵 ⊆ 𝑥))
9		df-ral 3062	. . 3 ⊢ (∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥) ↔ ∀𝑥(𝑥 ∈ Tarski → (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥)))
10	7, 8, 9	3bitr4i 302	. 2 ⊢ (𝐵 ⊆ ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)} ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥))
11	5, 10	bitrdi 286	1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1539   ∈ wcel 2106  {cab 2709  ∀wral 3061  {crab 3432   ⊆ wss 3948  ∩ cint 4950  ‘cfv 6543  Tarskictsk 10742  tarskiMapctskm 10831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-groth 10817

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-tsk 10743  df-tskm 10832

This theorem is referenced by: (None)