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Mirrors > Home > ILE Home > Th. List > fimacnvdisj | GIF version |
Description: The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.) |
Ref | Expression |
---|---|
fimacnvdisj | ⊢ ((𝐹:A⟶B ∧ (B ∩ 𝐶) = ∅) → (◡𝐹 “ 𝐶) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rn 4299 | . . . 4 ⊢ ran 𝐹 = dom ◡𝐹 | |
2 | frn 4995 | . . . . 5 ⊢ (𝐹:A⟶B → ran 𝐹 ⊆ B) | |
3 | 2 | adantr 261 | . . . 4 ⊢ ((𝐹:A⟶B ∧ (B ∩ 𝐶) = ∅) → ran 𝐹 ⊆ B) |
4 | 1, 3 | syl5eqssr 2984 | . . 3 ⊢ ((𝐹:A⟶B ∧ (B ∩ 𝐶) = ∅) → dom ◡𝐹 ⊆ B) |
5 | ssdisj 3271 | . . 3 ⊢ ((dom ◡𝐹 ⊆ B ∧ (B ∩ 𝐶) = ∅) → (dom ◡𝐹 ∩ 𝐶) = ∅) | |
6 | 4, 5 | sylancom 397 | . 2 ⊢ ((𝐹:A⟶B ∧ (B ∩ 𝐶) = ∅) → (dom ◡𝐹 ∩ 𝐶) = ∅) |
7 | imadisj 4630 | . 2 ⊢ ((◡𝐹 “ 𝐶) = ∅ ↔ (dom ◡𝐹 ∩ 𝐶) = ∅) | |
8 | 6, 7 | sylibr 137 | 1 ⊢ ((𝐹:A⟶B ∧ (B ∩ 𝐶) = ∅) → (◡𝐹 “ 𝐶) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∩ cin 2910 ⊆ wss 2911 ∅c0 3218 ◡ccnv 4287 dom cdm 4288 ran crn 4289 “ cima 4291 ⟶wf 4841 |
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-f 4849 |
This theorem is referenced by: (None) |
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