Theorem eltskm 10599

Description: Belonging to (tarskiMap‘𝐴). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)

Assertion

Ref	Expression
eltskm	⊢ (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 eltskm

Step	Hyp	Ref	Expression
1		tskmval 10595	. . 3 ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
2	1	eleq2d 2824	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ 𝐵 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥}))
3		elex 3450	. . . 4 ⊢ (𝐵 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} → 𝐵 ∈ V)
4	3	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} → 𝐵 ∈ V))
5		tskmid 10596	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ (tarskiMap‘𝐴))
6		tskmcl 10597	. . . . . 6 ⊢ (tarskiMap‘𝐴) ∈ Tarski
7		eleq2 2827	. . . . . . . 8 ⊢ (𝑥 = (tarskiMap‘𝐴) → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ (tarskiMap‘𝐴)))
8		eleq2 2827	. . . . . . . 8 ⊢ (𝑥 = (tarskiMap‘𝐴) → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ (tarskiMap‘𝐴)))
9	7, 8	imbi12d 345	. . . . . . 7 ⊢ (𝑥 = (tarskiMap‘𝐴) → ((𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥) ↔ (𝐴 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ (tarskiMap‘𝐴))))
10	9	rspcv 3557	. . . . . 6 ⊢ ((tarskiMap‘𝐴) ∈ Tarski → (∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥) → (𝐴 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ (tarskiMap‘𝐴))))
11	6, 10	ax-mp 5	. . . . 5 ⊢ (∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥) → (𝐴 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ (tarskiMap‘𝐴)))
12	5, 11	syl5com 31	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥) → 𝐵 ∈ (tarskiMap‘𝐴)))
13		elex 3450	. . . 4 ⊢ (𝐵 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ V)
14	12, 13	syl6 35	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥) → 𝐵 ∈ V))
15		elintrabg 4892	. . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥)))
16	15	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ V → (𝐵 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥))))
17	4, 14, 16	pm5.21ndd 381	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥)))
18	2, 17	bitrd 278	1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068  Vcvv 3432  ∩ cint 4879  ‘cfv 6433  Tarskictsk 10504  tarskiMapctskm 10593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-groth 10579

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-er 8498  df-en 8734  df-dom 8735  df-tsk 10505  df-tskm 10594

This theorem is referenced by: (None)