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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 4465 | . . 3 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
2 | 1 | eqeq1i 2096 | . 2 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ ran (𝐴 ↾ 𝐵) = ∅) |
3 | dm0rn0 4666 | . 2 ⊢ (dom (𝐴 ↾ 𝐵) = ∅ ↔ ran (𝐴 ↾ 𝐵) = ∅) | |
4 | dmres 4747 | . . . 4 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) | |
5 | incom 3193 | . . . 4 ⊢ (𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ 𝐵) | |
6 | 4, 5 | eqtri 2109 | . . 3 ⊢ dom (𝐴 ↾ 𝐵) = (dom 𝐴 ∩ 𝐵) |
7 | 6 | eqeq1i 2096 | . 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 1290 ∩ cin 2999 ∅c0 3287 dom cdm 4452 ran crn 4453 ↾ cres 4454 “ cima 4455 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-br 3852 df-opab 3906 df-xp 4458 df-cnv 4460 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 |
This theorem is referenced by: fnimadisj 5147 fnimaeq0 5148 fimacnvdisj 5208 |
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