Theorem resima2 4976

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 4672	. 2 ⊢ ((𝐴 ↾ 𝐶) “ 𝐵) = ran ((𝐴 ↾ 𝐶) ↾ 𝐵)
2		resres 4954	. . . 4 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵))
3	2	rneqi 4890	. . 3 ⊢ ran ((𝐴 ↾ 𝐶) ↾ 𝐵) = ran (𝐴 ↾ (𝐶 ∩ 𝐵))
4		df-ss 3166	. . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∩ 𝐶) = 𝐵)
5		incom 3351	. . . . . . . 8 ⊢ (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶)
6	5	a1i 9	. . . . . . 7 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶))
7	6	reseq2d 4942	. . . . . 6 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ (𝐵 ∩ 𝐶)))
8	7	rneqd 4891	. . . . 5 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → ran (𝐴 ↾ (𝐶 ∩ 𝐵)) = ran (𝐴 ↾ (𝐵 ∩ 𝐶)))
9		reseq2 4937	. . . . . . 7 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ↾ 𝐵))
10	9	rneqd 4891	. . . . . 6 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → ran (𝐴 ↾ (𝐵 ∩ 𝐶)) = ran (𝐴 ↾ 𝐵))
11		df-ima 4672	. . . . . 6 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
12	10, 11	eqtr4di 2244	. . . . 5 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → ran (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 “ 𝐵))
13	8, 12	eqtrd 2226	. . . 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 2238	. 2 ⊢ (𝐵 ⊆ 𝐶 → ran ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 “ 𝐵))
16	1, 15	eqtrid 2238	1 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) “ 𝐵) = (𝐴 “ 𝐵))

Syntax hints:   → wi 4   = wceq 1364   ∩ cin 3152   ⊆ wss 3153  ran crn 4660   ↾ cres 4661   “ cima 4662

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672

This theorem is referenced by:  cnptopresti  14406  cnptoprest  14407