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Mirrors > Home > ILE Home > Th. List > f1domg | GIF version |
Description: The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 4-Sep-2004.) |
Ref | Expression |
---|---|
f1domg | ⊢ (B ∈ 𝐶 → (𝐹:A–1-1→B → A ≼ B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1dmex 5685 | . . . . 5 ⊢ ((𝐹:A–1-1→B ∧ B ∈ 𝐶) → A ∈ V) | |
2 | f1f 5035 | . . . . . 6 ⊢ (𝐹:A–1-1→B → 𝐹:A⟶B) | |
3 | fex 5331 | . . . . . 6 ⊢ ((𝐹:A⟶B ∧ A ∈ V) → 𝐹 ∈ V) | |
4 | 2, 3 | sylan 267 | . . . . 5 ⊢ ((𝐹:A–1-1→B ∧ A ∈ V) → 𝐹 ∈ V) |
5 | 1, 4 | syldan 266 | . . . 4 ⊢ ((𝐹:A–1-1→B ∧ B ∈ 𝐶) → 𝐹 ∈ V) |
6 | 5 | expcom 109 | . . 3 ⊢ (B ∈ 𝐶 → (𝐹:A–1-1→B → 𝐹 ∈ V)) |
7 | f1eq1 5030 | . . . 4 ⊢ (f = 𝐹 → (f:A–1-1→B ↔ 𝐹:A–1-1→B)) | |
8 | 7 | spcegv 2635 | . . 3 ⊢ (𝐹 ∈ V → (𝐹:A–1-1→B → ∃f f:A–1-1→B)) |
9 | 6, 8 | syli 33 | . 2 ⊢ (B ∈ 𝐶 → (𝐹:A–1-1→B → ∃f f:A–1-1→B)) |
10 | brdomg 6165 | . 2 ⊢ (B ∈ 𝐶 → (A ≼ B ↔ ∃f f:A–1-1→B)) | |
11 | 9, 10 | sylibrd 158 | 1 ⊢ (B ∈ 𝐶 → (𝐹:A–1-1→B → A ≼ B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∃wex 1378 ∈ wcel 1390 Vcvv 2551 class class class wbr 3755 ⟶wf 4841 –1-1→wf1 4842 ≼ cdom 6156 |
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-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-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 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-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-dom 6159 |
This theorem is referenced by: f1dom 6176 dom2d 6189 |
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