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Mirrors > Home > ILE Home > Th. List > fnimaeq0 | GIF version |
Description: Images under a function never map nonempty sets to empty sets. (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 4630 | . 2 ⊢ ((𝐹 “ B) = ∅ ↔ (dom 𝐹 ∩ B) = ∅) | |
2 | incom 3123 | . . . 4 ⊢ (dom 𝐹 ∩ B) = (B ∩ dom 𝐹) | |
3 | fndm 4941 | . . . . . . 7 ⊢ (𝐹 Fn A → dom 𝐹 = A) | |
4 | 3 | sseq2d 2967 | . . . . . 6 ⊢ (𝐹 Fn A → (B ⊆ dom 𝐹 ↔ B ⊆ A)) |
5 | 4 | biimpar 281 | . . . . 5 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → B ⊆ dom 𝐹) |
6 | df-ss 2925 | . . . . 5 ⊢ (B ⊆ dom 𝐹 ↔ (B ∩ dom 𝐹) = B) | |
7 | 5, 6 | sylib 127 | . . . 4 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (B ∩ dom 𝐹) = B) |
8 | 2, 7 | syl5eq 2081 | . . 3 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (dom 𝐹 ∩ B) = B) |
9 | 8 | eqeq1d 2045 | . 2 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → ((dom 𝐹 ∩ B) = ∅ ↔ B = ∅)) |
10 | 1, 9 | syl5bb 181 | 1 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → ((𝐹 “ B) = ∅ ↔ B = ∅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∩ cin 2910 ⊆ wss 2911 ∅c0 3218 dom cdm 4288 “ cima 4291 Fn wfn 4840 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-cnv 4296 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-fn 4848 |
This theorem is referenced by: (None) |
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