Theorem tskmval 10696

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 10692	. . . . 5 ⊢ ∪ Tarski = V
3	1, 2	eleqtrrdi 2848	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ ∪ Tarski)
4		eluni2 4856	. . . 4 ⊢ (𝐴 ∈ ∪ Tarski ↔ ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥)
5	3, 4	sylib 217	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥)
6		intexrab 5284	. . 3 ⊢ (∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥 ↔ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ V)
7	5, 6	sylib 217	. 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ V)
8		eleq1 2824	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
9	8	rabbidv 3411	. . . 4 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ Tarski ∣ 𝑦 ∈ 𝑥} = {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
10	9	inteqd 4899	. . 3 ⊢ (𝑦 = 𝐴 → ∩ {𝑥 ∈ Tarski ∣ 𝑦 ∈ 𝑥} = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
11		df-tskm 10695	. . 3 ⊢ tarskiMap = (𝑦 ∈ V ↦ ∩ {𝑥 ∈ Tarski ∣ 𝑦 ∈ 𝑥})
12	10, 11	fvmptg 6929	. 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 2105  ∃wrex 3070  {crab 3403  Vcvv 3441  ∪ cuni 4852  ∩ cint 4894  ‘cfv 6479  Tarskictsk 10605  tarskiMapctskm 10694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372  ax-groth 10680

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-int 4895  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-iota 6431  df-fun 6481  df-fv 6487  df-tsk 10606  df-tskm 10695

This theorem is referenced by:  tskmid  10697  tskmcl  10698  sstskm  10699  eltskm  10700