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Mirrors > Home > MPE Home > Th. List > fnimadisj | Structured version Visualization version 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 6449 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
2 | 1 | ineq1d 4187 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∩ 𝐶) = (𝐴 ∩ 𝐶)) |
3 | 2 | eqeq1d 2823 | . . 3 ⊢ (𝐹 Fn 𝐴 → ((dom 𝐹 ∩ 𝐶) = ∅ ↔ (𝐴 ∩ 𝐶) = ∅)) |
4 | 3 | biimpar 480 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) = ∅) → (dom 𝐹 ∩ 𝐶) = ∅) |
5 | imadisj 5942 | . 2 ⊢ ((𝐹 “ 𝐶) = ∅ ↔ (dom 𝐹 ∩ 𝐶) = ∅) | |
6 | 4, 5 | sylibr 236 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 “ 𝐶) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∩ cin 3934 ∅c0 4290 dom cdm 5549 “ cima 5552 Fn wfn 6344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-cnv 5557 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-fn 6352 |
This theorem is referenced by: poimirlem15 34901 aacllem 44896 |
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