Theorem imaiinfv 42674

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 6623	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵)
2		fniinfv 6921	. . 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 6859	. . . 4 ⊢ (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
5	4	iineq2i 4974	. . 3 ⊢ ∩ 𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) = ∩ 𝑥 ∈ 𝐵 (𝐹‘𝑥)
6	5	eqcomi 2738	. 2 ⊢ ∩ 𝑥 ∈ 𝐵 (𝐹‘𝑥) = ∩ 𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥)
7		df-ima 5644	. . 3 ⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
8	7	inteqi 4910	. 2 ⊢ ∩ (𝐹 “ 𝐵) = ∩ ran (𝐹 ↾ 𝐵)
9	3, 6, 8	3eqtr4g 2789	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ∩ 𝑥 ∈ 𝐵 (𝐹‘𝑥) = ∩ (𝐹 “ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ⊆ wss 3911  ∩ cint 4906  ∩ ciin 4952  ran crn 5632   ↾ cres 5633   “ cima 5634   Fn wfn 6494  ‘cfv 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-fv 6507

This theorem is referenced by:  elrfirn  42676