Theorem resima2 4939

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 4638	. 2 ⊢ ((𝐴 ↾ 𝐶) “ 𝐵) = ran ((𝐴 ↾ 𝐶) ↾ 𝐵)
2		resres 4917	. . . 4 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵))
3	2	rneqi 4853	. . 3 ⊢ ran ((𝐴 ↾ 𝐶) ↾ 𝐵) = ran (𝐴 ↾ (𝐶 ∩ 𝐵))
4		df-ss 3142	. . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∩ 𝐶) = 𝐵)
5		incom 3327	. . . . . . . 8 ⊢ (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶)
6	5	a1i 9	. . . . . . 7 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶))
7	6	reseq2d 4905	. . . . . 6 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ (𝐵 ∩ 𝐶)))
8	7	rneqd 4854	. . . . 5 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → ran (𝐴 ↾ (𝐶 ∩ 𝐵)) = ran (𝐴 ↾ (𝐵 ∩ 𝐶)))
9		reseq2 4900	. . . . . . 7 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ↾ 𝐵))
10	9	rneqd 4854	. . . . . 6 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → ran (𝐴 ↾ (𝐵 ∩ 𝐶)) = ran (𝐴 ↾ 𝐵))
11		df-ima 4638	. . . . . 6 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
12	10, 11	eqtr4di 2228	. . . . 5 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → ran (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 “ 𝐵))
13	8, 12	eqtrd 2210	. . . 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 2222	. 2 ⊢ (𝐵 ⊆ 𝐶 → ran ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 “ 𝐵))
16	1, 15	eqtrid 2222	1 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) “ 𝐵) = (𝐴 “ 𝐵))

Syntax hints:   → wi 4   = wceq 1353   ∩ cin 3128   ⊆ wss 3129  ran crn 4626   ↾ cres 4627   “ cima 4628

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-xp 4631  df-rel 4632  df-cnv 4633  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638

This theorem is referenced by:  cnptopresti  13600  cnptoprest  13601