Theorem tskmval 10848

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 3471	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		grothtsk 10844	. . . . 5 ⊢ ∪ Tarski = V
3	1, 2	eleqtrrdi 2871	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ ∪ Tarski)
4		eluni2 4871	. . . 4 ⊢ (𝐴 ∈ ∪ Tarski ↔ ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥)
5	3, 4	sylib 221	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥)
6		intexrab 5311	. . 3 ⊢ (∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥 ↔ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ V)
7	5, 6	sylib 221	. 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ V)
8		eleq1 2848	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
9	8	rabbidv 3419	. . . 4 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ Tarski ∣ 𝑦 ∈ 𝑥} = {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
10	9	inteqd 4912	. . 3 ⊢ (𝑦 = 𝐴 → ∩ {𝑥 ∈ Tarski ∣ 𝑦 ∈ 𝑥} = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
11		df-tskm 10847	. . 3 ⊢ tarskiMap = (𝑦 ∈ V ↦ ∩ {𝑥 ∈ Tarski ∣ 𝑦 ∈ 𝑥})
12	10, 11	fvmptg 6984	. 2 ⊢ ((𝐴 ∈ V ∧ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ V) → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
13	1, 7, 12	syl2anc 596	1 ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  {crab 3412  Vcvv 3450  ∪ cuni 4867  ∩ cint 4907  ‘cfv 6533  Tarskictsk 10757  tarskiMapctskm 10846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pr 5398  ax-groth 10832

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-tsk 10758  df-tskm 10847

This theorem is used by:  tskmid  10849  tskmcl  10850  sstskm  10851  eltskm  10852