Theorem riinint 5382

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 4804	. . . . . . 7 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝑆 ∈ V)
2	1	expcom 451	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝑆 ⊆ 𝑋 → 𝑆 ∈ V))
3	2	ralimdv 2963	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋 → ∀𝑘 ∈ 𝐼 𝑆 ∈ V))
4	3	imp 445	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → ∀𝑘 ∈ 𝐼 𝑆 ∈ V)
5		dfiin3g 5379	. . . 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 3814	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) = (𝑋 ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆)))
8		intun 4509	. . 3 ⊢ ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) = (∩ {𝑋} ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆))
9		intsng 4512	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ∩ {𝑋} = 𝑋)
10	9	adantr 481	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → ∩ {𝑋} = 𝑋)
11	10	ineq1d 3813	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (∩ {𝑋} ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) = (𝑋 ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆)))
12	8, 11	syl5eq 2668	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) = (𝑋 ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆)))
13	7, 12	eqtr4d 2659	1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) = ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  {csn 4177  ∩ cint 4475  ∩ ciin 4521   ↦ cmpt 4729  ran crn 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-int 4476  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  cmpfiiin  37260