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| Mirrors > Home > MPE Home > Th. List > fnimaeq0 | Structured version Visualization version GIF version | ||
| Description: Images under a function never map nonempty sets to empty sets. EDITORIAL: usable in fnwe2lem2 43063. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
| Ref | Expression |
|---|---|
| fnimaeq0 | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝐹 “ 𝐵) = ∅ ↔ 𝐵 = ∅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | imadisj 6098 | . 2 ⊢ ((𝐹 “ 𝐵) = ∅ ↔ (dom 𝐹 ∩ 𝐵) = ∅) | |
| 2 | incom 4209 | . . . 4 ⊢ (dom 𝐹 ∩ 𝐵) = (𝐵 ∩ dom 𝐹) | |
| 3 | fndm 6671 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
| 4 | 3 | sseq2d 4016 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴)) |
| 5 | 4 | biimpar 477 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ dom 𝐹) |
| 6 | dfss2 3969 | . . . . 5 ⊢ (𝐵 ⊆ dom 𝐹 ↔ (𝐵 ∩ dom 𝐹) = 𝐵) | |
| 7 | 5, 6 | sylib 218 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐵 ∩ dom 𝐹) = 𝐵) |
| 8 | 2, 7 | eqtrid 2789 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (dom 𝐹 ∩ 𝐵) = 𝐵) |
| 9 | 8 | eqeq1d 2739 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((dom 𝐹 ∩ 𝐵) = ∅ ↔ 𝐵 = ∅)) |
| 10 | 1, 9 | bitrid 283 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝐹 “ 𝐵) = ∅ ↔ 𝐵 = ∅)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 dom cdm 5685 “ cima 5688 Fn wfn 6556 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fn 6564 |
| This theorem is referenced by: ipodrsima 18586 mdegldg 26105 ig1peu 26214 ig1pdvds 26219 dimval 33651 dimvalfi 33652 nummin 35105 aks6d1c6lem3 42173 kelac1 43075 |
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