Theorem resima2 5038

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 4731	. 2 ⊢ ((𝐴 ↾ 𝐶) “ 𝐵) = ran ((𝐴 ↾ 𝐶) ↾ 𝐵)
2		resres 5016	. . . 4 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵))
3	2	rneqi 4951	. . 3 ⊢ ran ((𝐴 ↾ 𝐶) ↾ 𝐵) = ran (𝐴 ↾ (𝐶 ∩ 𝐵))
4		df-ss 3210	. . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∩ 𝐶) = 𝐵)
5		incom 3396	. . . . . . . 8 ⊢ (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶)
6	5	a1i 9	. . . . . . 7 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶))
7	6	reseq2d 5004	. . . . . 6 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ (𝐵 ∩ 𝐶)))
8	7	rneqd 4952	. . . . 5 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → ran (𝐴 ↾ (𝐶 ∩ 𝐵)) = ran (𝐴 ↾ (𝐵 ∩ 𝐶)))
9		reseq2 4999	. . . . . . 7 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ↾ 𝐵))
10	9	rneqd 4952	. . . . . 6 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → ran (𝐴 ↾ (𝐵 ∩ 𝐶)) = ran (𝐴 ↾ 𝐵))
11		df-ima 4731	. . . . . 6 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
12	10, 11	eqtr4di 2280	. . . . 5 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → ran (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 “ 𝐵))
13	8, 12	eqtrd 2262	. . . 4 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → ran (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 “ 𝐵))
14	4, 13	sylbi 121	. . 3 ⊢ (𝐵 ⊆ 𝐶 → ran (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 “ 𝐵))
15	3, 14	eqtrid 2274	. 2 ⊢ (𝐵 ⊆ 𝐶 → ran ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 “ 𝐵))
16	1, 15	eqtrid 2274	1 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) “ 𝐵) = (𝐴 “ 𝐵))

Syntax hints:   → wi 4   = wceq 1395   ∩ cin 3196   ⊆ wss 3197  ran crn 4719   ↾ cres 4720   “ cima 4721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-xp 4724  df-rel 4725  df-cnv 4726  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731

This theorem is referenced by:  cnptopresti  14906  cnptoprest  14907