Theorem riinint 5894

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 5260	. . . . . . 7 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝑆 ∈ V)
2	1	expcom 414	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝑆 ⊆ 𝑋 → 𝑆 ∈ V))
3	2	ralimdv 3163	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋 → ∀𝑘 ∈ 𝐼 𝑆 ∈ V))
4	3	imp 407	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → ∀𝑘 ∈ 𝐼 𝑆 ∈ V)
5		dfiin3g 5891	. . . 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 4156	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) = (𝑋 ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆)))
8		intun 4922	. . 3 ⊢ ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) = (∩ {𝑋} ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆))
9		intsng 4927	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ∩ {𝑋} = 𝑋)
10	9	adantr 481	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → ∩ {𝑋} = 𝑋)
11	10	ineq1d 4155	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (∩ {𝑋} ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) = (𝑋 ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆)))
12	8, 11	eqtrid 2789	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) = (𝑋 ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆)))
13	7, 12	eqtr4d 2780	1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) = ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∀wral 3062  Vcvv 3441   ∪ cun 3894   ∩ cin 3895   ⊆ wss 3896  {csn 4569  ∩ cint 4890  ∩ ciin 4936   ↦ cmpt 5168  ran crn 5606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5236  ax-nul 5243  ax-pr 5365

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-int 4891  df-iin 4938  df-br 5086  df-opab 5148  df-mpt 5169  df-cnv 5613  df-dm 5615  df-rn 5616

This theorem is referenced by:  cmpfiiin  40729