Theorem eltskm 10842

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 10838	. . 3 ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
2	1	eleq2d 2817	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ 𝐵 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥}))
3		elex 3491	. . . 4 ⊢ (𝐵 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} → 𝐵 ∈ V)
4	3	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} → 𝐵 ∈ V))
5		tskmid 10839	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ (tarskiMap‘𝐴))
6		tskmcl 10840	. . . . . 6 ⊢ (tarskiMap‘𝐴) ∈ Tarski
7		eleq2 2820	. . . . . . . 8 ⊢ (𝑥 = (tarskiMap‘𝐴) → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ (tarskiMap‘𝐴)))
8		eleq2 2820	. . . . . . . 8 ⊢ (𝑥 = (tarskiMap‘𝐴) → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ (tarskiMap‘𝐴)))
9	7, 8	imbi12d 343	. . . . . . 7 ⊢ (𝑥 = (tarskiMap‘𝐴) → ((𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥) ↔ (𝐴 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ (tarskiMap‘𝐴))))
10	9	rspcv 3609	. . . . . 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 3491	. . . 4 ⊢ (𝐵 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ V)
14	12, 13	syl6 35	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥) → 𝐵 ∈ V))
15		elintrabg 4966	. . . 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 378	. 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 2104  ∀wral 3059  {crab 3430  Vcvv 3472  ∩ cint 4951  ‘cfv 6544  Tarskictsk 10747  tarskiMapctskm 10836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729  ax-groth 10822

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-er 8707  df-en 8944  df-dom 8945  df-tsk 10748  df-tskm 10837

This theorem is referenced by: (None)