Theorem resiima 6050

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 5654	. . 3 ⊢ (( I ↾ 𝐴) “ 𝐵) = ran (( I ↾ 𝐴) ↾ 𝐵)
2	1	a1i 11	. 2 ⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐵) = ran (( I ↾ 𝐴) ↾ 𝐵))
3		resabs1 5980	. . 3 ⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) ↾ 𝐵) = ( I ↾ 𝐵))
4	3	rneqd 5905	. 2 ⊢ (𝐵 ⊆ 𝐴 → ran (( I ↾ 𝐴) ↾ 𝐵) = ran ( I ↾ 𝐵))
5		rnresi 6049	. . 3 ⊢ ran ( I ↾ 𝐵) = 𝐵
6	5	a1i 11	. 2 ⊢ (𝐵 ⊆ 𝐴 → ran ( I ↾ 𝐵) = 𝐵)
7	2, 4, 6	3eqtrd 2769	1 ⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵)

Syntax hints:   → wi 4   = wceq 1540   ⊆ wss 3917   I cid 5535  ran crn 5642   ↾ cres 5643   “ cima 5644

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654

This theorem is referenced by:  fipreima  9316  psgnunilem1  19430  islinds2  21729  lindsind2  21735  ssidcn  23149  idqtop  23600  fmid  23854  ellspds  33346  rrhre  34018  sitmcl  34349  bj-imdirid  37181  bj-iminvid  37190  poimirlem15  37636  grimidvtxedg  47889  ushggricedg  47931  imaidfu2lem  49102  imaidfu  49103  imaidfu2  49104