Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > resima2 | GIF version |
Description: Image under a restricted class. (Contributed by FL, 31-Aug-2009.) |
Ref | Expression |
---|---|
resima2 | ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) “ 𝐵) = (𝐴 “ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 4522 | . 2 ⊢ ((𝐴 ↾ 𝐶) “ 𝐵) = ran ((𝐴 ↾ 𝐶) ↾ 𝐵) | |
2 | resres 4801 | . . . 4 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵)) | |
3 | 2 | rneqi 4737 | . . 3 ⊢ ran ((𝐴 ↾ 𝐶) ↾ 𝐵) = ran (𝐴 ↾ (𝐶 ∩ 𝐵)) |
4 | df-ss 3054 | . . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∩ 𝐶) = 𝐵) | |
5 | incom 3238 | . . . . . . . 8 ⊢ (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶) | |
6 | 5 | a1i 9 | . . . . . . 7 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶)) |
7 | 6 | reseq2d 4789 | . . . . . 6 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ (𝐵 ∩ 𝐶))) |
8 | 7 | rneqd 4738 | . . . . 5 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → ran (𝐴 ↾ (𝐶 ∩ 𝐵)) = ran (𝐴 ↾ (𝐵 ∩ 𝐶))) |
9 | reseq2 4784 | . . . . . . 7 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ↾ 𝐵)) | |
10 | 9 | rneqd 4738 | . . . . . 6 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → ran (𝐴 ↾ (𝐵 ∩ 𝐶)) = ran (𝐴 ↾ 𝐵)) |
11 | df-ima 4522 | . . . . . 6 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
12 | 10, 11 | syl6eqr 2168 | . . . . 5 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → ran (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 “ 𝐵)) |
13 | 8, 12 | eqtrd 2150 | . . . 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 | syl5eq 2162 | . 2 ⊢ (𝐵 ⊆ 𝐶 → ran ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 “ 𝐵)) |
16 | 1, 15 | syl5eq 2162 | 1 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) “ 𝐵) = (𝐴 “ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∩ cin 3040 ⊆ wss 3041 ran crn 4510 ↾ cres 4511 “ cima 4512 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-cnv 4517 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 |
This theorem is referenced by: cnptopresti 12334 cnptoprest 12335 |
Copyright terms: Public domain | W3C validator |