Theorem sstskm 10834

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 10831	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
2		df-rab 3425	. . . . 5 ⊢ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} = {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)}
3	2	inteqi 4945	. . . 4 ⊢ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} = ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)}
4	1, 3	eqtrdi 2780	. . 3 ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)})
5	4	sseq2d 4007	. 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 1813	. . 3 ⊢ (∀𝑥((𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥) → 𝐵 ⊆ 𝑥) ↔ ∀𝑥(𝑥 ∈ Tarski → (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥)))
8		ssintab 4960	. . 3 ⊢ (𝐵 ⊆ ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)} ↔ ∀𝑥((𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥) → 𝐵 ⊆ 𝑥))
9		df-ral 3054	. . 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 205   ∧ wa 395  ∀wal 1531   ∈ wcel 2098  {cab 2701  ∀wral 3053  {crab 3424   ⊆ wss 3941  ∩ cint 4941  ‘cfv 6534  Tarskictsk 10740  tarskiMapctskm 10829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-groth 10815

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-tsk 10741  df-tskm 10830

This theorem is referenced by: (None)