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Mirrors > Home > ILE Home > Th. List > fnimadisj | GIF version |
Description: A class that is disjoint with the domain of a function has an empty image under the function. (Contributed by FL, 24-Jan-2007.) |
Ref | Expression |
---|---|
fnimadisj | ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 “ 𝐶) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fndm 5230 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
2 | 1 | ineq1d 3281 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∩ 𝐶) = (𝐴 ∩ 𝐶)) |
3 | 2 | eqeq1d 2149 | . . 3 ⊢ (𝐹 Fn 𝐴 → ((dom 𝐹 ∩ 𝐶) = ∅ ↔ (𝐴 ∩ 𝐶) = ∅)) |
4 | 3 | biimpar 295 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) = ∅) → (dom 𝐹 ∩ 𝐶) = ∅) |
5 | imadisj 4909 | . 2 ⊢ ((𝐹 “ 𝐶) = ∅ ↔ (dom 𝐹 ∩ 𝐶) = ∅) | |
6 | 4, 5 | sylibr 133 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 “ 𝐶) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∩ cin 3075 ∅c0 3368 dom cdm 4547 “ cima 4550 Fn wfn 5126 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-cnv 4555 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-fn 5134 |
This theorem is referenced by: (None) |
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