Theorem riinint 5985

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 5329	. . . . . . 7 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝑆 ∈ V)
2	1	expcom 413	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝑆 ⊆ 𝑋 → 𝑆 ∈ V))
3	2	ralimdv 3167	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋 → ∀𝑘 ∈ 𝐼 𝑆 ∈ V))
4	3	imp 406	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → ∀𝑘 ∈ 𝐼 𝑆 ∈ V)
5		dfiin3g 5982	. . . 4 ⊢ (∀𝑘 ∈ 𝐼 𝑆 ∈ V → ∩ 𝑘 ∈ 𝐼 𝑆 = ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆))
6	4, 5	syl 17	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → ∩ 𝑘 ∈ 𝐼 𝑆 = ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆))
7	6	ineq2d 4228	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) = (𝑋 ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆)))
8		intun 4985	. . 3 ⊢ ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) = (∩ {𝑋} ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆))
9		intsng 4988	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ∩ {𝑋} = 𝑋)
10	9	adantr 480	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → ∩ {𝑋} = 𝑋)
11	10	ineq1d 4227	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (∩ {𝑋} ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) = (𝑋 ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆)))
12	8, 11	eqtrid 2787	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) = (𝑋 ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆)))
13	7, 12	eqtr4d 2778	1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) = ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  Vcvv 3478   ∪ cun 3961   ∩ cin 3962   ⊆ wss 3963  {csn 4631  ∩ cint 4951  ∩ ciin 4997   ↦ cmpt 5231  ran crn 5690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-int 4952  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-cnv 5697  df-dm 5699  df-rn 5700

This theorem is referenced by:  cmpfiiin  42685