Theorem eltskm 10796

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 10792	. . 3 ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
2	1	eleq2d 2814	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ 𝐵 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥}))
3		elex 3468	. . . 4 ⊢ (𝐵 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} → 𝐵 ∈ V)
4	3	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} → 𝐵 ∈ V))
5		tskmid 10793	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ (tarskiMap‘𝐴))
6		tskmcl 10794	. . . . . 6 ⊢ (tarskiMap‘𝐴) ∈ Tarski
7		eleq2 2817	. . . . . . . 8 ⊢ (𝑥 = (tarskiMap‘𝐴) → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ (tarskiMap‘𝐴)))
8		eleq2 2817	. . . . . . . 8 ⊢ (𝑥 = (tarskiMap‘𝐴) → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ (tarskiMap‘𝐴)))
9	7, 8	imbi12d 344	. . . . . . 7 ⊢ (𝑥 = (tarskiMap‘𝐴) → ((𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥) ↔ (𝐴 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ (tarskiMap‘𝐴))))
10	9	rspcv 3584	. . . . . 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 3468	. . . 4 ⊢ (𝐵 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ V)
14	12, 13	syl6 35	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥) → 𝐵 ∈ V))
15		elintrabg 4925	. . . 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 379	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥)))
18	2, 17	bitrd 279	1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3405  Vcvv 3447  ∩ cint 4910  ‘cfv 6511  Tarskictsk 10701  tarskiMapctskm 10790

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-groth 10776

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-er 8671  df-en 8919  df-dom 8920  df-tsk 10702  df-tskm 10791

This theorem is referenced by: (None)