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Mirrors > Home > MPE Home > Th. List > dom3d | Structured version Visualization version 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 9013 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵) |
4 | f1f 6793 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵) |
6 | dom3d.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
7 | dom3d.4 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
8 | fex2 7941 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ V) | |
9 | 5, 6, 7, 8 | syl3anc 1369 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ V) |
10 | f1eq1 6788 | . . 3 ⊢ (𝑧 = (𝑥 ∈ 𝐴 ↦ 𝐶) → (𝑧:𝐴–1-1→𝐵 ↔ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵)) | |
11 | 9, 3, 10 | spcedv 3585 | . 2 ⊢ (𝜑 → ∃𝑧 𝑧:𝐴–1-1→𝐵) |
12 | brdomg 8977 | . . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ≼ 𝐵 ↔ ∃𝑧 𝑧:𝐴–1-1→𝐵)) | |
13 | 7, 12 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 ≼ 𝐵 ↔ ∃𝑧 𝑧:𝐴–1-1→𝐵)) |
14 | 11, 13 | mpbird 257 | 1 ⊢ (𝜑 → 𝐴 ≼ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 Vcvv 3471 class class class wbr 5148 ↦ cmpt 5231 ⟶wf 6544 –1-1→wf1 6545 ≼ cdom 8962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fv 6556 df-dom 8966 |
This theorem is referenced by: dom3 9017 xpdom2 9092 fopwdom 9105 |
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