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Mirrors > Home > ILE Home > Th. List > imadisj | GIF version |
Description: A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.) |
Ref | Expression |
---|---|
imadisj | ⊢ ((𝐴 “ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 4522 | . . 3 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
2 | 1 | eqeq1i 2125 | . 2 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ ran (𝐴 ↾ 𝐵) = ∅) |
3 | dm0rn0 4726 | . 2 ⊢ (dom (𝐴 ↾ 𝐵) = ∅ ↔ ran (𝐴 ↾ 𝐵) = ∅) | |
4 | dmres 4810 | . . . 4 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) | |
5 | incom 3238 | . . . 4 ⊢ (𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ 𝐵) | |
6 | 4, 5 | eqtri 2138 | . . 3 ⊢ dom (𝐴 ↾ 𝐵) = (dom 𝐴 ∩ 𝐵) |
7 | 6 | eqeq1i 2125 | . 2 ⊢ (dom (𝐴 ↾ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅) |
8 | 2, 3, 7 | 3bitr2i 207 | 1 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1316 ∩ cin 3040 ∅c0 3333 dom cdm 4509 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-in1 588 ax-in2 589 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-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-cnv 4517 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 |
This theorem is referenced by: fnimadisj 5213 fnimaeq0 5214 fimacnvdisj 5277 |
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