Theorem resima2 4925

Description: Image under a restricted class. (Contributed by FL, 31-Aug-2009.)

Assertion

Ref	Expression
resima2	⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) “ 𝐵) = (𝐴 “ 𝐵))

Proof of Theorem resima2

Step	Hyp	Ref	Expression
1		df-ima 4624	. 2 ⊢ ((𝐴 ↾ 𝐶) “ 𝐵) = ran ((𝐴 ↾ 𝐶) ↾ 𝐵)
2		resres 4903	. . . 4 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵))
3	2	rneqi 4839	. . 3 ⊢ ran ((𝐴 ↾ 𝐶) ↾ 𝐵) = ran (𝐴 ↾ (𝐶 ∩ 𝐵))
4		df-ss 3134	. . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∩ 𝐶) = 𝐵)
5		incom 3319	. . . . . . . 8 ⊢ (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶)
6	5	a1i 9	. . . . . . 7 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶))
7	6	reseq2d 4891	. . . . . 6 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ (𝐵 ∩ 𝐶)))
8	7	rneqd 4840	. . . . 5 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → ran (𝐴 ↾ (𝐶 ∩ 𝐵)) = ran (𝐴 ↾ (𝐵 ∩ 𝐶)))
9		reseq2 4886	. . . . . . 7 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ↾ 𝐵))
10	9	rneqd 4840	. . . . . 6 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → ran (𝐴 ↾ (𝐵 ∩ 𝐶)) = ran (𝐴 ↾ 𝐵))
11		df-ima 4624	. . . . . 6 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
12	10, 11	eqtr4di 2221	. . . . 5 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → ran (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 “ 𝐵))
13	8, 12	eqtrd 2203	. . . 4 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → ran (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 “ 𝐵))
14	4, 13	sylbi 120	. . 3 ⊢ (𝐵 ⊆ 𝐶 → ran (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 “ 𝐵))
15	3, 14	eqtrid 2215	. 2 ⊢ (𝐵 ⊆ 𝐶 → ran ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 “ 𝐵))
16	1, 15	eqtrid 2215	1 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) “ 𝐵) = (𝐴 “ 𝐵))

Syntax hints:   → wi 4   = wceq 1348   ∩ cin 3120   ⊆ wss 3121  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-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-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:  cnptopresti  13032  cnptoprest  13033