Theorem resiima 4969

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 4624	. . 3 ⊢ (( I ↾ 𝐴) “ 𝐵) = ran (( I ↾ 𝐴) ↾ 𝐵)
2	1	a1i 9	. 2 ⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐵) = ran (( I ↾ 𝐴) ↾ 𝐵))
3		resabs1 4920	. . 3 ⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) ↾ 𝐵) = ( I ↾ 𝐵))
4	3	rneqd 4840	. 2 ⊢ (𝐵 ⊆ 𝐴 → ran (( I ↾ 𝐴) ↾ 𝐵) = ran ( I ↾ 𝐵))
5		rnresi 4968	. . 3 ⊢ ran ( I ↾ 𝐵) = 𝐵
6	5	a1i 9	. 2 ⊢ (𝐵 ⊆ 𝐴 → ran ( I ↾ 𝐵) = 𝐵)
7	2, 4, 6	3eqtrd 2207	1 ⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵)

Syntax hints:   → wi 4   = wceq 1348   ⊆ wss 3121   I cid 4273  ran crn 4612   ↾ cres 4613   “ cima 4614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624

This theorem is referenced by:  ssidcn  13004