Theorem tskmval 10870

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 3492	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		grothtsk 10866	. . . . 5 ⊢ ∪ Tarski = V
3	1, 2	eleqtrrdi 2840	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ ∪ Tarski)
4		eluni2 4916	. . . 4 ⊢ (𝐴 ∈ ∪ Tarski ↔ ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥)
5	3, 4	sylib 217	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥)
6		intexrab 5346	. . 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 2817	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
9	8	rabbidv 3438	. . . 4 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ Tarski ∣ 𝑦 ∈ 𝑥} = {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
10	9	inteqd 4958	. . 3 ⊢ (𝑦 = 𝐴 → ∩ {𝑥 ∈ Tarski ∣ 𝑦 ∈ 𝑥} = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
11		df-tskm 10869	. . 3 ⊢ tarskiMap = (𝑦 ∈ V ↦ ∩ {𝑥 ∈ Tarski ∣ 𝑦 ∈ 𝑥})
12	10, 11	fvmptg 7008	. 2 ⊢ ((𝐴 ∈ V ∧ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ V) → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
13	1, 7, 12	syl2anc 582	1 ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  ∃wrex 3067  {crab 3430  Vcvv 3473  ∪ cuni 4912  ∩ cint 4953  ‘cfv 6553  Tarskictsk 10779  tarskiMapctskm 10868

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-groth 10854

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-tsk 10780  df-tskm 10869

This theorem is referenced by:  tskmid  10871  tskmcl  10872  sstskm  10873  eltskm  10874