Theorem eltskm 10063

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 10059	. . 3 ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
2	1	eleq2d 2851	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ 𝐵 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥}))
3		elex 3433	. . . 4 ⊢ (𝐵 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} → 𝐵 ∈ V)
4	3	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} → 𝐵 ∈ V))
5		tskmid 10060	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ (tarskiMap‘𝐴))
6		tskmcl 10061	. . . . . 6 ⊢ (tarskiMap‘𝐴) ∈ Tarski
7		eleq2 2854	. . . . . . . 8 ⊢ (𝑥 = (tarskiMap‘𝐴) → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ (tarskiMap‘𝐴)))
8		eleq2 2854	. . . . . . . 8 ⊢ (𝑥 = (tarskiMap‘𝐴) → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ (tarskiMap‘𝐴)))
9	7, 8	imbi12d 337	. . . . . . 7 ⊢ (𝑥 = (tarskiMap‘𝐴) → ((𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥) ↔ (𝐴 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ (tarskiMap‘𝐴))))
10	9	rspcv 3531	. . . . . 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 3433	. . . 4 ⊢ (𝐵 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ V)
14	12, 13	syl6 35	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥) → 𝐵 ∈ V))
15		elintrabg 4762	. . . 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 372	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥)))
18	2, 17	bitrd 271	1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1507   ∈ wcel 2050  ∀wral 3088  {crab 3092  Vcvv 3415  ∩ cint 4749  ‘cfv 6188  Tarskictsk 9968  tarskiMapctskm 10057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-groth 10043

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-int 4750  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-er 8089  df-en 8307  df-dom 8308  df-tsk 9969  df-tskm 10058

This theorem is referenced by: (None)