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Theorem eltskm 9703
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 9699 . . 3 (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
21eleq2d 2716 . 2 (𝐴𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ 𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥}))
3 elex 3243 . . . 4 (𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥} → 𝐵 ∈ V)
43a1i 11 . . 3 (𝐴𝑉 → (𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥} → 𝐵 ∈ V))
5 tskmid 9700 . . . . 5 (𝐴𝑉𝐴 ∈ (tarskiMap‘𝐴))
6 tskmcl 9701 . . . . . 6 (tarskiMap‘𝐴) ∈ Tarski
7 eleq2 2719 . . . . . . . 8 (𝑥 = (tarskiMap‘𝐴) → (𝐴𝑥𝐴 ∈ (tarskiMap‘𝐴)))
8 eleq2 2719 . . . . . . . 8 (𝑥 = (tarskiMap‘𝐴) → (𝐵𝑥𝐵 ∈ (tarskiMap‘𝐴)))
97, 8imbi12d 333 . . . . . . 7 (𝑥 = (tarskiMap‘𝐴) → ((𝐴𝑥𝐵𝑥) ↔ (𝐴 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ (tarskiMap‘𝐴))))
109rspcv 3336 . . . . . 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 3243 . . . 4 (𝐵 ∈ (tarskiMap‘𝐴) → 𝐵 ∈ V)
1412, 13syl6 35 . . 3 (𝐴𝑉 → (∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥) → 𝐵 ∈ V))
15 elintrabg 4521 . . . 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 368 . 2 (𝐴𝑉 → (𝐵 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥)))
182, 17bitrd 268 1 (𝐴𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231   cint 4507  cfv 5926  Tarskictsk 9608  tarskiMapctskm 9697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-groth 9683
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-er 7787  df-en 7998  df-dom 7999  df-tsk 9609  df-tskm 9698
This theorem is referenced by: (None)
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