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Mirrors > Home > ILE Home > Th. List > inimass | GIF version |
Description: The image of an intersection. (Contributed by Thierry Arnoux, 16-Dec-2017.) |
Ref | Expression |
---|---|
inimass | ⊢ ((𝐴 ∩ 𝐵) “ 𝐶) ⊆ ((𝐴 “ 𝐶) ∩ (𝐵 “ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnin 4995 | . 2 ⊢ ran ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶)) ⊆ (ran (𝐴 ↾ 𝐶) ∩ ran (𝐵 ↾ 𝐶)) | |
2 | df-ima 4599 | . . 3 ⊢ ((𝐴 ∩ 𝐵) “ 𝐶) = ran ((𝐴 ∩ 𝐵) ↾ 𝐶) | |
3 | resindir 4882 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶)) | |
4 | 3 | rneqi 4814 | . . 3 ⊢ ran ((𝐴 ∩ 𝐵) ↾ 𝐶) = ran ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶)) |
5 | 2, 4 | eqtri 2178 | . 2 ⊢ ((𝐴 ∩ 𝐵) “ 𝐶) = ran ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶)) |
6 | df-ima 4599 | . . 3 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶) | |
7 | df-ima 4599 | . . 3 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶) | |
8 | 6, 7 | ineq12i 3306 | . 2 ⊢ ((𝐴 “ 𝐶) ∩ (𝐵 “ 𝐶)) = (ran (𝐴 ↾ 𝐶) ∩ ran (𝐵 ↾ 𝐶)) |
9 | 1, 5, 8 | 3sstr4i 3169 | 1 ⊢ ((𝐴 ∩ 𝐵) “ 𝐶) ⊆ ((𝐴 “ 𝐶) ∩ (𝐵 “ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ∩ cin 3101 ⊆ wss 3102 ran crn 4587 ↾ cres 4588 “ cima 4589 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-br 3966 df-opab 4026 df-xp 4592 df-rel 4593 df-cnv 4594 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 |
This theorem is referenced by: (None) |
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