Theorem tskmval 10752

Description: Value of our tarski map. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)

Assertion

Ref	Expression
tskmval	⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem tskmval

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3459	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		grothtsk 10748	. . . . 5 ⊢ ∪ Tarski = V
3	1, 2	eleqtrrdi 2839	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ ∪ Tarski)
4		eluni2 4865	. . . 4 ⊢ (𝐴 ∈ ∪ Tarski ↔ ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥)
5	3, 4	sylib 218	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥)
6		intexrab 5289	. . 3 ⊢ (∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥 ↔ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ V)
7	5, 6	sylib 218	. 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ V)
8		eleq1 2816	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
9	8	rabbidv 3404	. . . 4 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ Tarski ∣ 𝑦 ∈ 𝑥} = {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
10	9	inteqd 4904	. . 3 ⊢ (𝑦 = 𝐴 → ∩ {𝑥 ∈ Tarski ∣ 𝑦 ∈ 𝑥} = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
11		df-tskm 10751	. . 3 ⊢ tarskiMap = (𝑦 ∈ V ↦ ∩ {𝑥 ∈ Tarski ∣ 𝑦 ∈ 𝑥})
12	10, 11	fvmptg 6932	. 2 ⊢ ((𝐴 ∈ V ∧ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ V) → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
13	1, 7, 12	syl2anc 584	1 ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  {crab 3396  Vcvv 3438  ∪ cuni 4861  ∩ cint 4899  ‘cfv 6486  Tarskictsk 10661  tarskiMapctskm 10750

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 5238  ax-nul 5248  ax-pr 5374  ax-groth 10736

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 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-tsk 10662  df-tskm 10751

This theorem is referenced by:  tskmid  10753  tskmcl  10754  sstskm  10755  eltskm  10756