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Theorem eltskm 10253
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 10249 . . 3 (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
21eleq2d 2895 . 2 (𝐴𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ 𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥}))
3 elex 3510 . . . 4 (𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥} → 𝐵 ∈ V)
43a1i 11 . . 3 (𝐴𝑉 → (𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥} → 𝐵 ∈ V))
5 tskmid 10250 . . . . 5 (𝐴𝑉𝐴 ∈ (tarskiMap‘𝐴))
6 tskmcl 10251 . . . . . 6 (tarskiMap‘𝐴) ∈ Tarski
7 eleq2 2898 . . . . . . . 8 (𝑥 = (tarskiMap‘𝐴) → (𝐴𝑥𝐴 ∈ (tarskiMap‘𝐴)))
8 eleq2 2898 . . . . . . . 8 (𝑥 = (tarskiMap‘𝐴) → (𝐵𝑥𝐵 ∈ (tarskiMap‘𝐴)))
97, 8imbi12d 346 . . . . . . 7 (𝑥 = (tarskiMap‘𝐴) → ((𝐴𝑥𝐵𝑥) ↔ (𝐴 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ (tarskiMap‘𝐴))))
109rspcv 3615 . . . . . 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 3510 . . . 4 (𝐵 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ V)
1412, 13syl6 35 . . 3 (𝐴𝑉 → (∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥) → 𝐵 ∈ V))
15 elintrabg 4880 . . . 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 381 . 2 (𝐴𝑉 → (𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥)))
182, 17bitrd 280 1 (𝐴𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wcel 2105  wral 3135  {crab 3139  Vcvv 3492   cint 4867  cfv 6348  Tarskictsk 10158  tarskiMapctskm 10247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-groth 10233
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-er 8278  df-en 8498  df-dom 8499  df-tsk 10159  df-tskm 10248
This theorem is referenced by: (None)
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