Theorem sstskm 10899

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 10896	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
2		df-rab 3413	. . . . 5 ⊢ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} = {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)}
3	2	inteqi 4910	. . . 4 ⊢ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} = ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)}
4	1, 3	eqtrdi 2811	. . 3 ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)})
5	4	sseq2d 3962	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ 𝐵 ⊆ ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)}))
6		impexp 456	. . . 4 ⊢ (((𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥) → 𝐵 ⊆ 𝑥) ↔ (𝑥 ∈ Tarski → (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥)))
7	6	albii 1852	. . 3 ⊢ (∀𝑥((𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥) → 𝐵 ⊆ 𝑥) ↔ ∀𝑥(𝑥 ∈ Tarski → (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥)))
8		ssintab 4924	. . 3 ⊢ (𝐵 ⊆ ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)} ↔ ∀𝑥((𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥) → 𝐵 ⊆ 𝑥))
9		df-ral 3077	. . 3 ⊢ (∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥) ↔ ∀𝑥(𝑥 ∈ Tarski → (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥)))
10	7, 8, 9	3bitr4i 306	. 2 ⊢ (𝐵 ⊆ ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)} ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥))
11	5, 10	bitrdi 290	1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   ∈ wcel 2145  {cab 2738  ∀wral 3076  {crab 3412   ⊆ wss 3898  ∩ cint 4906  ‘cfv 6527  Tarskictsk 10805  tarskiMapctskm 10894

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-groth 10880

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-iota 6483  df-fun 6529  df-fv 6535  df-tsk 10806  df-tskm 10895

This theorem is used by: (None)