Theorem imaiinfv 43484

Description: Indexed intersection of an image. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
imaiinfv	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ∩ 𝑥 ∈ 𝐵 (𝐹‘𝑥) = ∩ (𝐹 “ 𝐵))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐹

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem imaiinfv

Step	Hyp	Ref	Expression
1		fnssres 6662	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵)
2		fniinfv 6963	. . 3 ⊢ ((𝐹 ↾ 𝐵) Fn 𝐵 → ∩ 𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) = ∩ ran (𝐹 ↾ 𝐵))
3	1, 2	syl 18	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ∩ 𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) = ∩ ran (𝐹 ↾ 𝐵))
4		fvres 6904	. . . 4 ⊢ (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
5	4	iineq2i 4981	. . 3 ⊢ ∩ 𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) = ∩ 𝑥 ∈ 𝐵 (𝐹‘𝑥)
6	5	eqcomi 2774	. 2 ⊢ ∩ 𝑥 ∈ 𝐵 (𝐹‘𝑥) = ∩ 𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥)
7		df-ima 5676	. . 3 ⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
8	7	inteqi 4918	. 2 ⊢ ∩ (𝐹 “ 𝐵) = ∩ ran (𝐹 ↾ 𝐵)
9	3, 6, 8	3eqtr4g 2825	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ∩ 𝑥 ∈ 𝐵 (𝐹‘𝑥) = ∩ (𝐹 “ 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ⊆ wss 3906  ∩ cint 4914  ∩ ciin 4959  ran crn 5664   ↾ cres 5665   “ cima 5666   Fn wfn 6535  ‘cfv 6540

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-fv 6548

This theorem is used by:  elrfirn  43486