Theorem resiima 6031

Description: The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)

Assertion

Ref	Expression
resiima	⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵)

Proof of Theorem resiima

Step	Hyp	Ref	Expression
1		df-ima 5636	. . 3 ⊢ (( I ↾ 𝐴) “ 𝐵) = ran (( I ↾ 𝐴) ↾ 𝐵)
2	1	a1i 11	. 2 ⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐵) = ran (( I ↾ 𝐴) ↾ 𝐵))
3		resabs1 5961	. . 3 ⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) ↾ 𝐵) = ( I ↾ 𝐵))
4	3	rneqd 5884	. 2 ⊢ (𝐵 ⊆ 𝐴 → ran (( I ↾ 𝐴) ↾ 𝐵) = ran ( I ↾ 𝐵))
5		rnresi 6030	. . 3 ⊢ ran ( I ↾ 𝐵) = 𝐵
6	5	a1i 11	. 2 ⊢ (𝐵 ⊆ 𝐴 → ran ( I ↾ 𝐵) = 𝐵)
7	2, 4, 6	3eqtrd 2768	1 ⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵)

Syntax hints:   → wi 4   = wceq 1540   ⊆ wss 3905   I cid 5517  ran crn 5624   ↾ cres 5625   “ cima 5626

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-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

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-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636

This theorem is referenced by:  fipreima  9267  psgnunilem1  19390  islinds2  21738  lindsind2  21744  ssidcn  23158  idqtop  23609  fmid  23863  ellspds  33318  rrhre  33990  sitmcl  34321  bj-imdirid  37162  bj-iminvid  37171  poimirlem15  37617  grimidvtxedg  47873  ushggricedg  47915  imaidfu2lem  49098  imaidfu  49099  imaidfu2  49100