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Mirrors > Home > ILE Home > Th. List > dom3d | GIF version |
Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by Mario Carneiro, 20-May-2013.) |
Ref | Expression |
---|---|
dom2d.1 | ⊢ (φ → (x ∈ A → 𝐶 ∈ B)) |
dom2d.2 | ⊢ (φ → ((x ∈ A ∧ y ∈ A) → (𝐶 = 𝐷 ↔ x = y))) |
dom3d.3 | ⊢ (φ → A ∈ 𝑉) |
dom3d.4 | ⊢ (φ → B ∈ 𝑊) |
Ref | Expression |
---|---|
dom3d | ⊢ (φ → A ≼ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dom2d.1 | . . . . . 6 ⊢ (φ → (x ∈ A → 𝐶 ∈ B)) | |
2 | dom2d.2 | . . . . . 6 ⊢ (φ → ((x ∈ A ∧ y ∈ A) → (𝐶 = 𝐷 ↔ x = y))) | |
3 | 1, 2 | dom2lem 6188 | . . . . 5 ⊢ (φ → (x ∈ A ↦ 𝐶):A–1-1→B) |
4 | f1f 5035 | . . . . 5 ⊢ ((x ∈ A ↦ 𝐶):A–1-1→B → (x ∈ A ↦ 𝐶):A⟶B) | |
5 | 3, 4 | syl 14 | . . . 4 ⊢ (φ → (x ∈ A ↦ 𝐶):A⟶B) |
6 | dom3d.3 | . . . 4 ⊢ (φ → A ∈ 𝑉) | |
7 | dom3d.4 | . . . 4 ⊢ (φ → B ∈ 𝑊) | |
8 | fex2 5002 | . . . 4 ⊢ (((x ∈ A ↦ 𝐶):A⟶B ∧ A ∈ 𝑉 ∧ B ∈ 𝑊) → (x ∈ A ↦ 𝐶) ∈ V) | |
9 | 5, 6, 7, 8 | syl3anc 1134 | . . 3 ⊢ (φ → (x ∈ A ↦ 𝐶) ∈ V) |
10 | f1eq1 5030 | . . . 4 ⊢ (z = (x ∈ A ↦ 𝐶) → (z:A–1-1→B ↔ (x ∈ A ↦ 𝐶):A–1-1→B)) | |
11 | 10 | spcegv 2635 | . . 3 ⊢ ((x ∈ A ↦ 𝐶) ∈ V → ((x ∈ A ↦ 𝐶):A–1-1→B → ∃z z:A–1-1→B)) |
12 | 9, 3, 11 | sylc 56 | . 2 ⊢ (φ → ∃z z:A–1-1→B) |
13 | brdomg 6165 | . . 3 ⊢ (B ∈ 𝑊 → (A ≼ B ↔ ∃z z:A–1-1→B)) | |
14 | 7, 13 | syl 14 | . 2 ⊢ (φ → (A ≼ B ↔ ∃z z:A–1-1→B)) |
15 | 12, 14 | mpbird 156 | 1 ⊢ (φ → A ≼ B) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∃wex 1378 ∈ wcel 1390 Vcvv 2551 class class class wbr 3755 ↦ cmpt 3809 ⟶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-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-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-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-fv 4853 df-dom 6159 |
This theorem is referenced by: dom3 6192 xpdom2 6241 fopwdom 6246 |
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