Theorem riinint 5954

Description: Express a relative indexed intersection as an intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
riinint	⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) = ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)))

Distinct variable groups:   𝑘,𝑉   𝑘,𝑋

Allowed substitution hints:   𝑆(𝑘)   𝐼(𝑘)

Proof of Theorem riinint

Step	Hyp	Ref	Expression
1		ssexg 5281	. . . . . . 7 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝑆 ∈ V)
2	1	expcom 419	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝑆 ⊆ 𝑋 → 𝑆 ∈ V))
3	2	ralimdv 3177	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋 → ∀𝑘 ∈ 𝐼 𝑆 ∈ V))
4	3	imp 412	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → ∀𝑘 ∈ 𝐼 𝑆 ∈ V)
5		dfiin3g 5951	. . . 4 ⊢ (∀𝑘 ∈ 𝐼 𝑆 ∈ V → ∩ 𝑘 ∈ 𝐼 𝑆 = ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆))
6	4, 5	syl 18	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → ∩ 𝑘 ∈ 𝐼 𝑆 = ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆))
7	6	ineq2d 4166	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) = (𝑋 ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆)))
8		intun 4940	. . 3 ⊢ ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) = (∩ {𝑋} ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆))
9		intsng 4943	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ∩ {𝑋} = 𝑋)
10	9	adantr 486	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → ∩ {𝑋} = 𝑋)
11	10	ineq1d 4165	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (∩ {𝑋} ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) = (𝑋 ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆)))
12	8, 11	eqtrid 2808	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) = (𝑋 ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆)))
13	7, 12	eqtr4d 2799	1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) = ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  {csn 4584  ∩ cint 4907  ∩ ciin 4952   ↦ cmpt 5186  ran crn 5652

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 2733  ax-sep 5249  ax-pr 5391

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-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-int 4908  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-cnv 5659  df-dm 5661  df-rn 5662

This theorem is used by:  cmpfiiin  43707