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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 (𝐴𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem eltskm
StepHypRef Expression
1 tskmval 10059 . . 3 (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
21eleq2d 2851 . 2 (𝐴𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ 𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥}))
3 elex 3433 . . . 4 (𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥} → 𝐵 ∈ V)
43a1i 11 . . 3 (𝐴𝑉 → (𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥} → 𝐵 ∈ V))
5 tskmid 10060 . . . . 5 (𝐴𝑉𝐴 ∈ (tarskiMap‘𝐴))
6 tskmcl 10061 . . . . . 6 (tarskiMap‘𝐴) ∈ Tarski
7 eleq2 2854 . . . . . . . 8 (𝑥 = (tarskiMap‘𝐴) → (𝐴𝑥𝐴 ∈ (tarskiMap‘𝐴)))
8 eleq2 2854 . . . . . . . 8 (𝑥 = (tarskiMap‘𝐴) → (𝐵𝑥𝐵 ∈ (tarskiMap‘𝐴)))
97, 8imbi12d 337 . . . . . . 7 (𝑥 = (tarskiMap‘𝐴) → ((𝐴𝑥𝐵𝑥) ↔ (𝐴 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ (tarskiMap‘𝐴))))
109rspcv 3531 . . . . . 6 ((tarskiMap‘𝐴) ∈ Tarski → (∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥) → (𝐴 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ (tarskiMap‘𝐴))))
116, 10ax-mp 5 . . . . 5 (∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥) → (𝐴 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ (tarskiMap‘𝐴)))
125, 11syl5com 31 . . . 4 (𝐴𝑉 → (∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥) → 𝐵 ∈ (tarskiMap‘𝐴)))
13 elex 3433 . . . 4 (𝐵 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ V)
1412, 13syl6 35 . . 3 (𝐴𝑉 → (∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥) → 𝐵 ∈ V))
15 elintrabg 4762 . . . 4 (𝐵 ∈ V → (𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥)))
1615a1i 11 . . 3 (𝐴𝑉 → (𝐵 ∈ V → (𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥))))
174, 14, 16pm5.21ndd 372 . 2 (𝐴𝑉 → (𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥)))
182, 17bitrd 271 1 (𝐴𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥)))
Colors of variables: wff setvar class
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)
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