Theorem imaiinfv 42796

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 6604	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵)
2		fniinfv 6900	. . 3 ⊢ ((𝐹 ↾ 𝐵) Fn 𝐵 → ∩ 𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) = ∩ ran (𝐹 ↾ 𝐵))
3	1, 2	syl 17	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ∩ 𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) = ∩ ran (𝐹 ↾ 𝐵))
4		fvres 6841	. . . 4 ⊢ (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
5	4	iineq2i 4962	. . 3 ⊢ ∩ 𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) = ∩ 𝑥 ∈ 𝐵 (𝐹‘𝑥)
6	5	eqcomi 2740	. 2 ⊢ ∩ 𝑥 ∈ 𝐵 (𝐹‘𝑥) = ∩ 𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥)
7		df-ima 5627	. . 3 ⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
8	7	inteqi 4899	. 2 ⊢ ∩ (𝐹 “ 𝐵) = ∩ ran (𝐹 ↾ 𝐵)
9	3, 6, 8	3eqtr4g 2791	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ∩ 𝑥 ∈ 𝐵 (𝐹‘𝑥) = ∩ (𝐹 “ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ⊆ wss 3897  ∩ cint 4895  ∩ ciin 4940  ran crn 5615   ↾ cres 5616   “ cima 5617   Fn wfn 6476  ‘cfv 6481

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-fv 6489

This theorem is referenced by:  elrfirn  42798