Theorem sstskm 10725

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 10722	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
2		df-rab 3394	. . . . 5 ⊢ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} = {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)}
3	2	inteqi 4899	. . . 4 ⊢ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} = ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)}
4	1, 3	eqtrdi 2781	. . 3 ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)})
5	4	sseq2d 3965	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ 𝐵 ⊆ ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)}))
6		impexp 450	. . . 4 ⊢ (((𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥) → 𝐵 ⊆ 𝑥) ↔ (𝑥 ∈ Tarski → (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥)))
7	6	albii 1820	. . 3 ⊢ (∀𝑥((𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥) → 𝐵 ⊆ 𝑥) ↔ ∀𝑥(𝑥 ∈ Tarski → (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥)))
8		ssintab 4913	. . 3 ⊢ (𝐵 ⊆ ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)} ↔ ∀𝑥((𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥) → 𝐵 ⊆ 𝑥))
9		df-ral 3046	. . 3 ⊢ (∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥) ↔ ∀𝑥(𝑥 ∈ Tarski → (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥)))
10	7, 8, 9	3bitr4i 303	. 2 ⊢ (𝐵 ⊆ ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)} ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥))
11	5, 10	bitrdi 287	1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   ∈ wcel 2110  {cab 2708  ∀wral 3045  {crab 3393   ⊆ wss 3900  ∩ cint 4895  ‘cfv 6477  Tarskictsk 10631  tarskiMapctskm 10720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-groth 10706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6433  df-fun 6479  df-fv 6485  df-tsk 10632  df-tskm 10721

This theorem is referenced by: (None)